MECHANICS  DEPT 


Engineering 
Library 


MACHINE    DESIGN 


BY 


ALBERT    W.    SMITH 

Director  of  Sibley  College,  Cornell  University 

(RETIRED) 
AND 

GUIDO   H.   MARX 

Professor  of  Machine  Design 
Leland  Stanford  Junior  University 


FOURTH  EDITION,  REVISED  AND  ENLARGED 
TOTAL  ISSUE,  EIGHT  THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 
LONDON:   CHAPMAN  &  HALL,  LIMITED 


:  •  ; 
•  ••• 


Engineering 
Library 


Copyright,  1905,  1908,  1909,  1915,  1920,  1922. 

BY 
ALBERT  W.  SMITH  AND   GUIDO  H.  MARX 


2/22 


PRESS    OF 

BRAUNWORTH    &    CO. 

BOOK    MANUFACTURERS 

BROOKLYN.    N.    Y. 


PREFACE    TO    FOURTH    EDITION. 


THE  call  for  a  new  edition  of  this  book  has  given  opportunity 
for  thorough  revision  of  the  text  and  the  inclusion  of  results  of 
recent  investigations  of  machine  elements.  The  task,  by  cordial 
mutual  consent,  was  undertaken  and  has  been  executed  solely 
by  the  junior  author,  upon  whom  the  entire  responsibility  for  the 
book  now  rests. 

The  original  plan  of  emphasizing  fundamental  principles 
and  methods  of  reasoning  is  retained  and  there  has  been  no 
effort  to  make  the  volume  cyclopaedic  in  scope. 

Engineering  literature  has  been  freely  consulted  and  the  in- 
debtedness to  other  writers  is  acknowledged  in  the  text,  as  in  the 
earlier  editions.  Similarly,  references  are  made  to  more  exhaust- 
ive treatments  of  many  topics  than  are  possible  in  this  volume. 

Thanks  are  extended  to  Assistant  Professor  L.  E.  Cutter, 
Mr.  B.  M.  Green,  graduate  assistant  in  machine  design,  and  Mr. 
D.  J.  Conant,  all  of  Stanford  University,  for  the  preparation  of 
many  of  the  drawings  for  the  new  illustrations. 

iii 


6092T6 


PREFACE   TO   THE    SECOND   EDITION. 


ONE  can  never  become  a  machine  designer  by  studying  books. 
Much  help  may  come  from  books,  but  the  true  designer  must 
have  judgment,  ripened  by  experience,  in  constructing  and 
operating  machines.  One  may  know  the  laws  that  govern  the 
development,  transmission  and  application  of  energy;  may  have 
knowledge  of  constructive  materials;  may  know  how  to  obtain 
results  by  mathematical  processes,  and  yet  be  unable  to  design 
a  good  machine.  There  is  also  needed  a  knowledge  of  many 
things  connected  with  manufacture,  transportation,  erection  and 
operation.  With  this  knowledge  it  is  possible  to  take  results 
of  computation  and  accept,  reject  and  modify  until  a  machine 
is  produced  that  will  do  the  required  work  satisfactorily. 

Professor  John  E.  Sweet  once  said,  "  It  is  comparatively  easy 
to  design  a  good  new  machine,  but  it  is  very  hard  to  design  a 
machine  that  will  be  good  when  it  is  old."  A  machine  must 
not  only  do  its  work  at  first,  but  must  continue  to  do  it  with  a 
minimum  of  repairs  as  long  as  the  work  needs  to  be  done.  The 
designer  must  be  able  to  foresee  the  results  of  machine  operation; 
he  must  have  imagination.  This  is  an  inborn  power,  but  it  may 
be  developed  by  use  and  by  engineering  experience. 

But  there  is  a  certain  part  of  the  designer's  mental  equipment 
that  may  be  furnished  in  the  class-room,  or  by  books.  This  is 
the  excuse  for  the  following  pages.  Machine  design  cannot  be 
treated  exhaustively.  There  are  too  many  kinds  of  machines 
for  this  and  their  differences  are  too  great.  In  this  book  an 


vi  PREFACE. 


effort  is  made  simply  to  give  principles  that  underlie  all  machine 
design  and  to  suggest  methods  of  reasoning  which  may  be  helpful 
in  the  designing  of  any  machine.  A  knowledge  of  the  usual 
university  course  in  pure  and  applied  mathematics  is  pre- 
supposed. 


CONTENTS. 


CHAPTER  I. 

PAGE 

PRELIMINARY   ......................................................       I 


CHAPTER  II. 
MOTION  IN  MECHANISMS  ____  ........................................     15 

CHAPTER   III. 
PARALLEL  OR  STRAIGHT-LINE  MOTIONS  .........................  .  ......    41 

CHAPTER  IV. 
CAMS  ..............  ..........  .............................  ........    5i 

CHAPTER  V. 
ENERGY  IN  MACHINES   ...........................  ."  .................    62 

CHAPTER  VI. 
PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS  ...............    81 

CHAPTER  VII. 
RIVETED  JOINTS   ...................  .........  ...................  ,  .  .  .  105 

CHAPTER  VIII. 
BOLTS  AND  SCREWS  .................  ........  .......................   139 

CHAPTER  IX. 

MEANS  FOR  PREVENTING  RELATIVE  ROTATION  ...................  .....   168 

vii 


viii  CONTENTS. 

CHAPTER   X. 

PAGE 

SLIDING  SURFACES 185 

CHAPTER   XI. 
AXLES,  SHAFTS,  AND  SPINDLES 194 

CHAPTER   XII. 
JOURNALS,  BEARINGS,  AND  LUBRICATION 210 

CHAPTER  XIII, 
ROLLER-  AND  BALL-BEARINGS 261 

CHAPTER  XIV. 
COUPLINGS  AND  CLUTCHES 274 

CHAPTER  XV. 
BELTS,  ROPES,  BRAKES,  AND  CHAINS 286 

CHAPTER  XVI. 
FLY-WHEELS  AND  PULLEYS 328 

CHAPTER  XVII. 
TOOTHED  WHEELS  OR  GEARS 347 

CHAPTER  XVIII. 
SPRINGS   •  •  429 

CHAPTER  XIX. 
MACHINE  SUPPORTS 436 

CHAPTER   XX. 
MACHINE  FRAMES    .  • 441 

APPENDIX 477 

INDEX   485 


INTRODUCTION. 


IN  general  there  are  five  considerations  of  prime  importance 
in  designing  machines:  I.  Adaptation,  II.  Strength  and  Stiff- 
ness, III.  Economy,  IV.  Appearance,  V.  Safety. 

I.  This  requires  all  complexity  to  be  reduced  to  its  lowest 
terms  in  order  that  the  machine  shall  accomplish  the  desired 
result  in  the  most  direct  way  possible,  and  with  greatest  convenience 
to  the  operator. 

II.  This  requires  the  machine  parts  subjected  to  the  action  of 
forces  to  sustain  these  forces,  not  only  without  rupture,  but  also 
without  such  yielding  as  would  interfere  with  the  accurate  action 
of  the  machine.     In  many  cases  the  forces  to  be  resisted  may 
be  calculated,  and  the  laws  of  mechanics  and  the  known  qualities 
of  constructive  materials  become  factors  in  determining  propor- 
tions.    In  other  cases  the  force,  by  the  use  of  a  "breaking-piece," 
may  be  limited  to  a  maximum  value,  which  therefore  dictates 
the  design.     But  in  many  other  cases  the  forces  acting  are  neces- 
sarily unknown;    and  appeal  must  be  made  to  the  precedent  of 
successful  practice,  or  to  the  judgment  of  some  experienced  man, 
until  one's  own  judgment  becomes  trustworthy  by  experience. 

In  proportioning  machine  parts,  the  designer  must  always  be 
sure  that  the  stress  which  is  the  basis  of  the  calculation  or  the 
estimate,  is  the  maximum  possible  stress;  otherwise  the  part 
will  be  incorrectly  proportioned.  For  instance,  if  the  arms  of  a 

pulley  were  to  be  designed  solely  on  the  assumption  that  they 

ix 


x  INTRODUCTION. 

endure  only  the  transverse  stress  due  to  the  belt  tension,  they 
would  be  found  to  be  absurdly  small,  because  the  stresses  resulting 
from  the  shrinkage  of  the  casting  in  cooling  are  often  far  greater 
than  those  due  to  the  belt  pull. 

The  design  of  many  machines  is  a  result  of  what  may  be  called 
"machine  evolution."  The  first  machine  was  built  according  to 
the  best  judgment  of  its  designer;  but  that  judgment  was  fallible, 
and  some  part  ruptured  under  the  stresses  sustained;  it  was  re- 
placed by  a  new  part  made  stronger;  it  ruptured  again,  and  again 
was  enlarged,  or  perhaps  made  of  some  more  suitable  material; 
it  then  sustained  the  applied  stresses  satisfactorily.  Some  other 
part  yielded  too  much  under  stress,  although  it  was  entirely  safe 
from  actual  rupture ;  this  part  was  then  stiffened  and  the  process 
continued  till  the  whole  machine  became  properly  proportioned 
for  the  resisting  of  stress.  Many  valuable  lessons  have  been  learned 
from  this  process;  many  excellent  machines  have  resulted  from 
it.  There  are,  however,  two  objections  to  it:  it  is  slow  and  very 
expensive,  and  if  any  part  had  originally  an  excess  of  material, 
it  is  not  changed;  only  the  parts  that  yield  are  perfected. 

Modern  analytical  methods  are  rightly  displacing  it  in  all 
progressive  establishments. 

III.  The  attainment  of  economy  does  not  necessarily  mean  the 
saving  of  metal  or  labor,  although  it  may  mean  that.  To  illustrate : 
Suppose  that  it  is  required  to  design  an  engine -lathe  for  the 
market.  The  competition  is  sharp;  the  profits  are  small.  How 
shall  the  designer  change  the  design  of  the  lathes  on  the  market 
to  increase  profits?  (a)  He  may,  if  possible,  reduce  the  weight 
of  metal  used,  maintaining  strength  and  stiffness  by  better  dis- 
tribution. But  this  must  not  increase  labor  in  the  foundry  or 
machine-shop,  nor  reduce  weight  which  prevents  undue  vibrations. 
(b)  He  may  design  special  tools  to  reduce  labor  without  reduction 
of  the  standard  of  workmanship.  The  interest  on  the  first  cost 
of  these  special  tools,  however,  must  not  exceed  the  possible  gain 


INTRODUCTION.  xi 

from  increased  profits.  (c)  He  may  make  the  lathe  more  con- 
venient for  the  workmen.  True  economy  permits  some  increase 
in  cost  to  gain  this  end.  It  is  not  meant  that  elaborate  and 
expensive  devices  are  to  be  used,  such  as  often  come  from  men 
of  more  inventiveness  than  judgment;  but  that  if  the  parts  can 
be  rearranged,  or  in  any  way  changed,  so  that  the  lathes-man 
shall  select  this  lathe  to  use  because  it  is  handier  when  other 
lathes  are  available,  then  economy  has  been  served,  even  though 
the  cost  has  been  somewhat  increased,  because  the  favorable 
opinion  of  intelligent  workmen  means  increased  sales. 

In  (a)  economy  is  served  by  a  reduction  of  metal;  in  (b)  by  a 
reduction  of  labor;  in  (c)  it  may  be  served  by  an  increase  of  both 
labor  and  material. 

The  addition  of  material  largely  in  excess  of  that  necessary 
for  strength  and  rigidity,  to  reduce  vibrations,  may  also  be  in  the 
interest  of  economy,  because  it  may  increase  the  durability  of  the 
machine  and  its  foundation,  or  may  reduce  the  expense  incident 
upon  repairs  and  delays,  thereby  bettering  the  reputation  of  the 
machine  and  increasing  sales. 

Suppose,  to  illustrate  further,  that  a  machine  part  is  to  be 
designed,  and  either  of  two  forms,  A  or  5,  will  serve  equally  well. 
The  part  is  to  be  of  cast  iron.  The  pattern  for  A  will  cost  twice 
as  much  as  for  B.  In  the  foundry  and  machine-shop,  however, 
A  can  be  produced  a  very  little  cheaper  than  B-  Clearly  then  if 
but  one. machine  is  to  be  built,  B  should  be  decided  on;  whereas, 
if  the  machine  is  to  be  manufactured  in  large  numbers,  A  is 
preferable.  Expense  for  patterns  is  a  first  cost.  Expense  for 
work  in  the  foundry  and  machine-shop  is  repeated  with  each 
machine. 

Economy  of  operation  also  needs  attention.  This  depends 
upon  the  efficiency  of  the  machine ;  i.e.,  upon  the  proportion  of  the 
energy  supplied  to  the  machine  which  really  does  useful  work. 
This  efficiency  is  increased  by  the  reduction  of  useless  frictional 


xii  INTRODUCTION. 

resistances,  by  careful  attention  to  the  design  and  means  of  lubri- 
cation of  rubbing  surfaces. 

In  order  that  economy  may  be  best  attained,  the  machine 
designer  needs  to  be  familiar  with  all  the  processes  used  in  the 
construction  of  machines — pattern -making,  foundry  work,  forging, 
and  the  processes  of  the  machine-shop — and  must  have  them  con- 
stantly in  mind,  so  that  while  each  part  designed  is  made  strong 
enough  and  stiff  enough,  and  properly  and  conveniently  arranged, 
and  of  such  form  as  to  be  satisfactory  in  appearance,  it  also  is 
so  designed  that  the  cost  of  construction  is  a  minimum. 

IV.  The    fourth    important    consideration    is     Appearance. 
There  is  a  beauty  possible  of  attainment  in  the  design  of  machines 
which  is  always  the  outgrowth  of  a  purpose.    Otherwise  expressed, 
a  machine  to  be  beautiful  must  be  purposeful.     Ornament  for 
ornament's  sake  is  seldom  admissible  in  machine  design.     And 
yet  the  striving  for  a  pleasing  effect  is  as  much  a  part  of  the  duty 
of  a  machine  designer  as  it  is  a  part  of  the  duty  of  an  architect. 

As  a  guiding  principle,  the  general  rule  may  be  laid  down 
that  simplicity  and  directness  are  always  best.  Each  member 
should  be  studied  with  strict  reference  to  the  function  which  it 
is  to  perform  and  the  stresses  to  which  it  is  subjected  and  then 
given  the  form  and  size  best  suited  to  meet  the  conditions  with 
the  greatest  economy  of  material  and  workmanship.  When 
combined,  the  parts  must  be  modified  in  such  manner  as  may  be 
found  necessary^to  the  harmonious  effect  of  the  whole. 

V.  Safety  of  the   operator  and  others  who  come  into  the 
vicinity  of  the  machine  is  the  fifth  important  point  in  design. 
It  is  really  a  sub-division  of  Adaptation  but,  for  emphasis,  may 
be  given  the  prominence  of  a  separate  head.     Beyond  the  pro- 
visions for  Strength  and  Stiffness,   it  requires  that  all  moving 
parts  shall  be  so  formed  and  guarded  as  to  eliminate,  so  far  as 
may  be  foreseen,  all  danger  of  bodily  accident. 


MACHINE    DESIGN. 


CHAPTER  I. 

PRELIMINARY. 

i.  Definitions. — The  study  of  machine  design  is  based  upon 
the  science  of  mechanics,  which  treats  questions  involving  the 
consideration  of  motion,  force,  work,  and  energy.  Since  it  will 
be  necessary  to  use  these  terms  almost  continually,  it  is  well  to 
make  an  exact  statement  of  what  is  to  be  understood  by  them. 

Motion  may  be  defined  as  change  of  position  in  space. 

A  Force  is  one  of  a  pair  of  equal,  opposite,  and  simultaneous 
actions  between  two  bodies  by  which  the  state  of  their  motion 
is  altered,  or  a  change  in  the  form  or  condition  of  the  bodies  them- 
selves is  effected. 

Work  is  the  name  given  to  the  result  of  a  force  in  motion. 

Energy  is  the  capacity  possessed  by  matter  to  do  work. 

A  Machine  is  a  combination  of  resistant  bodies  whose  relative 
motions  are  completely  constrained,  and  whose  function  it  is  to 
transform  available  energy  into  useful  work. 

The  law  of  Conservation  of  Energy  underlies  every  machine 
problem.  This  law  may  be  expressed  as  follows:  The  sum  of 
energy  in  the  universe  is  constant.  Energy  may  be  transferred 
in  space;  it  may  be  stored  for  varying  lengths  of  time;  it  may 
be  changed  from  one  of  its  several  forms  to  another;  but  it  can- 
not be  created  or  destroyed. 


^  ^UJA-y 

2  MACHINE  DESIGN. 

The  application  of  this  law  to  machines  is  as  follows:  A 
machine  receives  energy  from  a  source,  and  uses  it  to  do  useful 
and  useless  work. 

A  single  cycle  of  action  of  a  machine  is  that  sequence  of  opera- 
tions during  which  each  member  of  the  machine  has  gone  once 
through  all  the  relative  motions  possible  to  it.  A  complete  cycle 
of  action  is  such  a  period  that  all  conditions  (velocities,  etc.,  as 
well  as  relative  positions)  in  the  machine  are  the  same  at  its 
beginning  and  end. 

During  a  single  cycle  of  action  of  the  machine,  the  energy 
received  equals  the  total  work  done.  The  work  done  may  appear 
as  (a)  useful  work  delivered  by  the  machine,  or  as  (b)  heat  due 
to  energy  transformed  through  frictional  resistance,  or  as  (c) 
stored  mechanical  energy  in  some  moving  part  of  the  machine 
whose  velocity  is  increased.  The  sign  of  the  stored  energy  may 
be  plus  or  minus,  so  that  energy  received  in  one  cycle  may  be 
delivered  during  another  cycle;  but  for  any  considerable  time 
interval  of  machine  action  the  algebraic  sum  of  the  stored  energy 
must  equal  zero. 

For  a  single  cycle: 
Energy  received  =  useful  work  +  useless  work  ± stored  energy. 

For  continuous  action: 

Energy  received  =  useful  work  +  useless  work. 
In  operation  a  machine  generally  acts  by  a  continuous  repetition 
of  its  cycle. 

2.  Efficiency  of  Machines. — In  general,  efficiency  may  be 
denned  as  the  ratio  of  a  result  to  the  effort  made  to  produce  that 
result.  In  a  machine  the  result  corresponds  to  the  useful  work, 
while  the  effort  corresponds  to  the  energy  received.  Hence  the 
efficiency  of  a  machine  =  useful  work  -j-  energy  received.*  The 
designer  must  strive  for  high  efficiency,  i.e.,  for  the  greatest 
possible  result  for  a  given  effort. 

*  The  work  and  energy  must,  of  course,  be  expressed  in  the  same  units. 


PRELIMINARY.  3 

3.  Function  of  Machines.  —  Nature  furnishes  sources  of 
energy,  and  the  supplying  of  human  needs  requires  work  to  be 
done.  The  function  of  machines  is  to  cause  matter  possessing 
energy  to  do  useful  work. 

The  chief  sources  of  energy  in  nature  available  for  machine 
purposes  are: 

ist.  The  energy  of  air  in  motion  (i.e.,  wind)  due  to  its  mass 
and  velocity. 

2d.  The  energy  of  water  due  to  its  mass  and  motion  or  posi- 
tion. 

3d.  The  energy  dormant  in  fuels  which  manifests  itself  as 
heat  upon  combustion. 

The  general  method  by  which  the  machine  function  is  exer- 
cised may  be  shown  by  the  following  illustration: 

Illustration. — The  water  in  a  mill-pond  possesses  energy 
(potential)  by  virtue  of  its  position.  The  earth  exerts  an  attrac- 
tive force  upon  it.  If  there  is  no  outlet,  the  earth's  attractive 
force  cannot  cause  motion;  and  hence,  since  motion  is  a  neces- 
sary factor  of  work,  no  work  is  done.. 

If  the  water  overflows  the  dam,  the  earth's  attraction  causes 
that  part  of  it  which  overflows  to  move  to  a  lower  level,  and  before 
it  can  be  brought  to  rest  again  it  does  work  against  the  force 
which  brings  it  to  rest.  If  this  water  simply  falls  upon  rocks, 
its  energy  is  transformed  into  heat,  with  no  useful  result. 

But  if  the  water  is  led  from  the  pond  to  a  lower  level,  in  a 
closed  pipe  which  connects  with  a  water-wheel,  it  will  act  upon 
the  vanes  of  the  wheel  (because  of  the  earth's  attraction),  and 
will  cause  the  wheel  and  its  shaft  to  rotate  against  resistance, 
whereby  it  may  do  useful  work.  The  water-wheel  is  a  machine 
and  is  called  a  Prime  Mover,  because  it  is  the  first  link  in  the 
machine-chain  between  natural  energy  and  useful  work,  and  it 
is  the  function  of  prime  movers  to  change  some  form  of  natural 
energy  into  a  form  applicable  to  the  performance  of  useful  work. 


4  MACHINE   DESIGN. 

Since  it  is  usually  necessary  to  do  the  required  work  at  some 
distance  from  the  necessary  location  of  the  water-wheel,  Machinery 
of  Transmission  is  used  (shafts,  pulleys,  belts,  cables,  etc.),  and 
the  rotative  energy  is  rendered  available  at  the  required  place.* 

But  this  rotative  energy  may  not  be  suitable  to  do  the  re- 
quired work;  the  rotation  may  be  too  slow  or  too  fast;  a  resist- 
ance may  need  to  be  overcome  in  straight,  parallel  lines,  or  at 
periodical  intervals.  Hence  Machinery  of  Application  is  intro- 
duced to  transform  the  energy  to  meet  the  requirements  of  the 
work  to  be  done.  Thus  the  chain  is  complete,  and  the  potential 
energy  of  the  water  does  the  required  useful  work. 

The  chain  of  machines  which  has  the  steam-boiler  and  engine 

for  its  prime  mover  transforms  the    potential    heat  energy  of 

fuel  into  useful  work.     This  might  be  analyzed  in  a  similar  way. 

4.  Free  Motion. — The  general  science  of  mechanics  treats  of 

the  action  of  forces  upon  "free  bodies." 

In  the  case  of  a  "  free  body  "  acted  on  by  a  system  of  forces 
not  in  equilibrium,  motion  results  in  the  direction  of  the  resultant 
of  the  system.  If  another  force  is  introduced  whose  line  of 
action  does  not  coincide  with  that  of  the  resultant,  the  line  of 
action  of  the  resultant  is  changed,  and  the  body  moves  in  a  new 
direction.  The  character  of  the  motion,  therefore,  is  dependent 
upon  the  forces  which  produce  the  motion. 
This  is  called  free  motion. 

Example. — In  Fig.  i,  suppose  the  free 
body  M  to  be  acted  on  by  the  concurrent 
forces  i,  2,  and  3  whose  lines  of  action 
pass  through  the  center  of  gravity  of  M. 
The  line  of  action  of  the  resultant  of  these 
forces  is  AB,  and  the  body's  center  of 
gravity  would  move  along  this  line. 

*  Electric,  hydraulic,  and  pneumatic  transmission  systems  are  also  em- 
ployed. 


PRELIMINARY.  5 

If  another  force,  4,  is  introduced,  CD  becomes  the  line  of 
action  of  the  resultant,  and  the  motion  of  the  body  is  along 
the  line  CD. 

5.  Constrained     Motion.  —  In    a   machine    certain   definite 
motions  occur;    any  departure  from  these  motions,  or  the  pro- 
duction  of    any   other   motions,    would    result    in    derangement 
of  the  action  of  the  machine.     Thus,  the  spindle  of  an  engine- 
lathe   turns   accurately   about   its    axis;    the   cutting-tool   moves 
parallel  to  the  spindle's  axis;  and  an  accurate  cylindrical  surface 
is  thereby  produced.     If  there  were  any  departure  from  these 
motions,  the  lathe  would  fail  to  do  its  required  work.     In  all 
machines  certain  definite  motions  must  be  produced,   and  all 
other  motions  must  be  prevented;    or,  in  other  words,  motion 
in  machines  must  be  constrained. 

Constrained  motion  differs  from  free  motion  in  being  inde- 
pendent of  the  forces  which  produce  it.  If  any  force,  not  suffi- 
ciently great  to  produce  deformation,  be  applied  to  a  body  whose 
motion  is  constrained,  the  result  is  either  a  certain  predeter- 
mined motion,  or  no  motion  at  all. 

6.  Force  Opposed  by  Passive  Resistance. — A  force  may  act 
without  being  able  to  produce  motion  (and  hence  without  being 
able  to  do  work),  as  in  the  case  of  the  water  in  a  mill-pond  without 
overflow  or  outlet.     This  may  be  further  illustrated:    Suppose  a 
force,  say  hand  pressure,  to  be  applied  vertically  to  the  top  of  a 
table.     The  material  of  the  table  offers  a  passive  resistance,  and 
the  force  is  unable  to  produce  motion,  or  to  do  work. 

It  is  therefore  possible  to  offer  passive  resistance  to  such 
forces  as  may  be  required  not  to  produce  motion,  thereby  render- 
ing them  incapable  of  doing  work.  Whenever  a  body  opposes 
a  passive  resistance  to  the  action  of  a  force  a  change  in  its  condi- 
tion is  effected:  the  force  sets  up  an  equivalent  stress  in  the 
material  of  the  body.  Thus,  when  the  table  offers  a  passive 
resistance  to  the  hand -pressure,  compressive  stress  is  induced 


6  MACHINE  DESIGN. 

in  the  legs.  In  every  case  the  material  of  the  body  must  be  of  such 
shape  and  strength  as  to  resist  successfully  the  induced  stress. 

In  a  machine  there  must  be  provision  for  resisting  every  pos- 
sible force  which  tends  to  produce  any  but  the  required  motion. 
This  provision  is  usually  made  by  means  of  the  passive  resistance 
of  properly  formed  and  sufficiently  resistant  metallic  surfaces. 

Illustration  I. — Fig.  2  represents  a  section  and  end  view  of 
a  wood-lathe  headstock.  It  is  required  that  the  spindle,  S,  and 
the  attached  cone  pulley,  C,  shall  have  no  other  motion  than 

-Q 


FIG.  2. 


rotation  about  the  axis  of  the  spindle.  If  any  other  motion  is 
possible,  this  machine  part  cannot  be  used  for  the  required  pur- 
pose. At  A  and  B  the  cylindrical  surfaces  of  the  spindle  are 
enclosed  by  accurately  fitted  bearings  or  internal  cylindrical  sur- 
faces. Suppose  any  force,  P,  whose  line  of  action  lies  in  the 
plane  of  the  paper,  to  be  applied  to  the  cone  pulley.  It  may  be 
resolved  into  a  radial  component,  R,  and  a  tangential  component, 
T.  The  passive  resistance  of  the  cylindrical  surfaces  of  the 
journal  and  its  bearing,  prevents  R  from  producing  motion; 
while  it  offers  no  resistance,  friction  being  disregarded,  to  the 
action  of  T,  which  is  allowed  to  produce  the  required  motion, 
i.e.,  rotation  about  the  spindle's  axis.  If  the  line  of  action  of  P 
pass  through  the  axis,  its  tangential  component  becomes  zero, 


PRELIMINARY.  7 

and  no  motion  results.  If  the  line  of  action  of  P  become  tangen- 
tial, its  radial  component  becomes  zero,  and  P  is  wholly  applied 
to  produce  rotation.  If  a  force  Q,  whose  line  of  action  lies  in 
the  plane  of  the  paper,  be  applied  to  the  cone,  it  may  be  resolved 
into  a  radial  component,  N,  and  a  component,  M,  parallel  to 
the  spindle's  axis.  N  is  resisted  as  before  by  the  journal  and 
bearing  surfaces,  and  M  is  resisted  by  the  shoulder  surfaces  of 
the  bearings^  which  fit  against  the  shoulder  surfaces  of  the  spindle 
collars.  The  force  Q  can  therefore  produce  no  motion  at  all. 

In  general,  any  force  applied  to  the  cone  pulley  may  be 
resolved  into  a  radial,  a  tangential,  and  an  axial  component. 
Of  these  only  the  tangential  component  is  able  to  produce  motion; 
and  that  motion  is  the  motion  required.  The  constrainment  is 
therefore  complete;  i.e.,  there  can  be  no  motion  except  rotation 
about  the  spindle's  axis.  This  result  is  due  to  the  passive  resist- 
ance of  metallic  surfaces. 

Illustration  II. — R,  Fig.  3,  represents,  with  all  details  omitted, 
the  "ram,"  or  portion  of  a  shaping-machine  which  carries  the 


FIG.  3. 

cutting-tool.  It  is  required  to  produce  plane  surfaces,  and  hence 
the  "ram"  must  have  accurate  rectilinear  motion  in  the  direction 
of  HK.  Any  deviation  from  such  motion  would  render  the 
machine  useless. 

Consider  Fig.  3,  A.  Any  force  which  can  be  applied  to  the 
ram  may  be  resolved  into  three  components:  one  vertical,  one 
horizontal  and  parallel  to  the  paper,  and  one  perpendicular  to 


8  MACHINE   DESIGN. 

the  paper.  The  vertical  component,  if  acting  upward,  is  resisted 
by  the  plane  surfaces  in  contact  at  C  and  D ;  if  acting  downward, 
it  is  resisted  by  the  plane  surfaces  in  contact  at  E.  Therefore 
no  vertical  component  can  produce  motion.  The  horizontal 
component  parallel  to  the  paper  is  resisted  by  the  plane  surfaces 
in  contact  at  F  or  G,  according  as  it  acts  toward  the  right  or 
left.  The  component  perpendicular  to  the  paper  is  free  to  pro- 
duce motion  in  the  direction  of  its  line  of  action;  but  this  is  the 
motion  required. 

Any  force,  therefore,  which  has  a  component  perpendicular 
to  the  paper  can  produce  the  required  motion,  but  no  other 
motion.  The  constrainment  is  therefore  complete,  and  the 
result  is  due  to  the  passive  resistance  offered  by  metallic  surfaces. 

Complete  Constrainment  is  not  always  required  in  machines. 
It  is  only  necessary  to  prevent  such  motions  as  interfere  with 
the  accomplishment  of  the  desired  result. 

The  weight  of  a  moving  part  is  sometimes  utilized  to  produce 
constrainment  in  one  direction.  Thus  in  a  planer-table,  and  in 
some  lathe-carriages,  downward  motion  and  unallowable  side 
motion  are  resisted  by  metallic  surfaces;  while  upward  motion 
is  resisted  by  the  weight  of  the  moving  part. 

From  the  foregoing  it  follows  that,  as  passive  resistances 
can  be  opposed  to  all  forces  whose  lines  of  action  do  not  coincide 
with  the  desired  direction  of  motion  of  any  machine  part,  it  may 
be  said  that  the  nature  of  the  motion  is  independent  of  the  forces 
producing  it. 

Since  the  motions  of  machine  parts  are  independent  oj  the  jorces 
producing  them,  it  follows  that  the  relation  oj  such  motions  may 
be  determined  without  bringing  force  into  the  consideration. 

7.  Kinds  of  Motion  in  Machines. — Motion  in  machines  may 
be  very  complex,  but  it  is  chiefly  plane  motion. 

When  a  body  moves  in  such  a  way  that  any  section  of  it  re- 
mains in  the  same  plane,  its  motion  is  called  plane  motion.  All 


PRELIMINARY.  9 

sections  parallel  to  the  above  section  must  also  remain,  each  in 
its  own  plane.  If  the  plane  motion  is  such  that  all  points  of  the 
moving  body  remain  at  a  constant  distance  from  some  line,  AB, 
the  motion  is  called  rotation  about  the  axis  AB.  Example. — 
A  line-shaft  with  attached  parts. 

If  all  points  of  a  body  move  in  straight  parallel  paths,  the 
motion  of  the  body  is  called  rectilinear  translation.  Examples. — 
Engine  cross-head,  lathe-carriage,  planer-table,  shaper-ram. 
Rectilinear  translation  may  be  conveniently  considered  as  a 
special  case  of  rotation,  in  which  the  axis  of  rotation  is  at  an 
infinite  distance,  at  right  angles  to  the  motion. 

If  a  body  moves  parallel  to  an  axis  about  which  it  rotates, 
the  body  is  said  to  have  helical  or  screw  motion.  Example.^ 
A  nut  turning  upon  a  station  aiy  screw. 

If  all  points  of  a  body,  whose  motion  is  not  plane  motion, 
move  so  that  their  distances  from  a  certain  point,  O,  remain 
constant,  the  motion  is  called  spheric  motion.  This  is  because 
each  point  moves  in  the  surface  of  a  sphere  whose  center  is  O. 
Example. — The  arms  of  a  fly-ball  steam-engine  governor,  when 
the  vertical  position  is  changing. 

8.  Relative  Motion. — The  motion  of  any  machine  part,  like 
all  known  motion,  is  relative  motion.  It  is  studied  by  reference 
to  some  other  part  of  the  same  machine.  Some  one  part  of  a 
machine  is  usually  (though  not  necessarily)  fixed,  i.e.,  it  has  no 
motion  relative  to  the  earth.  This  fixed  part  is  called  the  frame 
of  the  machine.  The  motion  of  a  machine  part  may  be  referred 
to  the  frame,  or,  as  is  often  necessary,  to  some  other  part  which 
also  has  motion  relative  to  the  frame. 

The  kind  and  amount  of  relative  motion  of  a  machine  part 
depend  upon  the  part  to  which  its  motion  is  referred.  Since  this 
is  so,  it  is  always  essential,  when  dealing  with  the  motion  of  a 
machine  part,  to  specify  clearly  the  standard  relative  to  which, 
for  the  time  being,  this  motion  is  being  considered. 


10 


MACHINE  DESIGN. 


A  is  the  frame;   C  is  a 


Illustration. — Fig.  4  shows  a  press. 
plate  which  is  so  constrained  that,  its 
motion  being  referred  to  A,  it  may 
move  vertically,  but  cannot  rotate. 
Motion  oi  rotation  is  communicated  to 
the  screw  B.  The  motion  of  B  re- 
ferred to  A  is  helical  motion,  i.e., 
combined  rotation  and  translation. 
C,  however,  shares  the  translation  of 
B,  and  hence  there  is  left  only  rotation 
as  the  relative  motion  of  B  and  C.  FIG.  4. 

The  motion  of  B  referred  to  C  is  rotation.     The  motion  of  C  re- 
ferred to  B  is  rotation.  The  motion  of  C  referred  to  A  is  translation. 

In  general,  if  two  machine  members,  M  and  N,  move  relative 
to  a  third  member,  R,  the  relative  motion  of  M  referred  to  N 
depends  on  how  much  of  the  motion  of  N  is  shared  by  M.  If 
M  and  N  have  the  same  motions  relative  to  R,  they  have  no  mo- 
tion relative  to  each  other. 

Conversely,  if  two  bodies  have  no  relative  motion,  they  have 
the  same  motion  relative  to  a  third  body.  Thus  in  Fig.  4,  if 
the  constrainment  of  C  were  such  that  it  could  share  B's  rotation, 
as  well  as  its  translation,  then  C  would  have  helical  motion  rela- 
tive to  the  frame,  and  no  motion  at  all  relative  to  B.  This  is 
assumed  to  be  self-evident. 

A  rigid  body  is  one  in  which  the  distance  between  elementary 
portions  *  is  constant.  No  body  is  absolutely  rigid,  but  usually 
in  machine  members  the  departure  from  rigidity  is  so  slight  that 
it  may  be  neglected. 

Many  machine  members,  as  springs,  etc.,  are  useful  because 
of  their  lack  of  rigidity. 


*  In  this  volume  this  term  is  used  as  interchangeable  with  the  term  "  par- 
ticles "  of  mechanics. 


PRELIMINARY.  II 

Points  *  in  a  rigid  body  can  have  no  relative  motion,  and 
hence  must  all  have  the  same  motion. 

9.  Instantaneous  Plane  Motion  and  Instantaneous  Centers 
or  Centres. — Points  of  a  moving  body  trace  more  or  less  complex 
paths.  If  a  point  be  considered  as 
moving  from  one  position  in  its  path  to 
another  indefinitely  near,  its  motion  is 
called  instantaneous  motion.  The  point 
is  moving,  for  the  instant,  along  a 
straight  line  joining  the  two  indefinitely 
near  together  positions,  and  such  a 
line  is  a  tangent  to  the  path.  In  problems  which  are  solved 
by  the  aid  of  the  conception  of  instantaneous  motion  it  is  only 
necessary  to  know  the  direction  of  motion;  hence,  for  such  pur- 
poses, the  instantaneous  motion  of  a  point  is  fully  defined  by  a 
tangent  to  its  path  at  the  position  occupied  by  the  point  at  the 
instant. 

Thus  in  Fig.  5,  if  a  point  is  moving  in  the  path  APB,  when 
it  occupies  the  position  P  the  tangent  TT  represents  its  instan- 
taneous motion.  Any  number  of  curves  could  be  drawn  tangent 
to  TT  at  P,  and  any  one  of  them  would  be  a  possible  path  of 
the  point;  but  whatever  path  it  is  following,  its  instantaneous 
motion  is  represented  by  TT.  The  instantaneous  motion  of  a 
point  is  therefore  independent  of  the  form  of  its  path.  Any  one 
of  the  possible  paths  may  be  considered  as  equivalent,  for  the 
instant,  to  a  circle  whose  center  is  anywhere  in  the  normal  NN. 

In  general,  the  instantaneous  motion  of  a  point,  P,  is  equiva- 
lent to  rotation  about  some  point,  O,  in  a  line  through  the  point  P 
perpendicular  to  the  direction  of  its  instantaneous  motion. 

Let  the  instantaneous  motion  of  a  point,  A,  Fig.  6,  in  a  sec- 
tion of  a  moving  body  be  given  by  the  line  TT.  Then  the  motion 

*  In  this  volume  this  term  is  used  as  interchangeable  with  the  term  "  par- 
ticles "  of  mechanics. 


12  MACHINE  DESIGN. 

is  equivalent  to  rotation  about  some  point  on  the  line  AB  as  a 

center,  but  it  may  be  any  point,  and  hence  the  instantaneous 

motion  of  the  body  is  not  determined.     But  if  the  instantaneous 

motion  of  another  point,  C,  be  given  by  the  line  T\T\,  this  motion 

is  equivalent  to  rotation  about  some  point  of  CD.     But  the  points 

A  and  C  are  points  in  a  rigid  body,  and  can  have  no  relative 

motion,  and  must  have  the  same  motion,  i.e.,  rotation  about  the 

same  center.     A  rotates  about  some  point  of  AB,  and  C  rotates 

about  some  point  of  CD;   but  they  must  rotate  about  the  same 

point,    and     the    only    point    which 

is  at   the   same   time   in   both   lines 

is    their    intersection,     O.         Hence 

A     and    C,     and    all    other    points 

of  the  body,  rotate,  for   the   instant, 

about    an    axis    of   which    O    is  the 

projection;    or,  in    other   words,   the 

instantaneous   motion    of    the    body  FlG   6 

is   rotation   about   an   axis  of  which 

O  is    the   projection.     This    axis    is   the   instantaneous   axis  of 

the    body's    motion,  and  O  is  the   instantaneous  center   of   the 

motion  of  the  section  shown  in  Fig.  6. 

For  the  sake  of  brevity  an  instantaneous  center  will  be  called 
a  centre. 

If  TT  and  TiTi  had  been  parallel  to  each  other,  AB  and 
CD  would  also  have  been  parallel,  and  would  have  intersected 
at  infinity;  in  which  case  the  body's  instantaneous  motion  would 
have  been  rotation  about  an  axis  infinitely  distant;  i.e.,  it  would 
have  been  translation. 

The  motion  of  the  body  in  Fig.  6  is  of  course  referred  to  a 
fixed  body,  which,  in  this  case,  may  be  represented  by  the  paper. 
The  instantaneous  motion  of  the  body  relative  to  the  paper  is 
rotation  about  O.  Let  M  represent  the  figure,  and  N  the  fixed 
body  represented  by  the  paper.  Suppose  the  material  of  M 


PRELIMINARY.  13 

to  be  extended  so  as  to  include  O.  Then  a  pin  could  be  put 
through  O,  materially  connecting  M  and  N,  without  interfering 
with  their  instantaneous  motion.  Such  connection  at  any  other 
point  would  interfere  with  the  instantaneous  motion. 

The  centra  of  the  relative  motion  of  two  bodies  is  a  point,  and 
the  only  one,  at  which  they  have  no  relative  motion;  it  is  a  point, 
and  the  only  one,  that  is  common  to  the  two  bodies  for  the  instant, 
and  which  may  be  considered  as  being  a  point  of  either;  it  is  a- 
point,  and  the  only  one,  about  which  as  center  either  body  may 
be  considered  as  rotating,  for  the  instant,  relative  to  the  other. 

It  will  be  seen  that  the  points  of  the  figure  in  Fig.  6  might 
be  moving  in  any  paths,  so  long  as  those  paths  are  tangent  at 
the  points  to  the  lines  representing  the  instantaneous  motion. 

In  general,  centres  of  the  relative  motion  of  two  bodies  are 
continually  changing  their  position.  They  may,  however,  remain 
stationary;  i.e.,  they  may  become  fixed  centers  of  rotation. 

10.  Loci  of  Centres,  or  Centrodes.* — As  centres  change  posi- 
tion they  describe  curves  of  some  kind,  and  these  loci  of  centres 
may  be  called  centrodes. 

Suppose  a  section  of  any  body,  M,  to  have  motion  relatively 
to  a  section  of  another  body,  N  (fixed),  in  the  same  or  a  parallel 
plane.  Centres  may  be  found  for  a  series  of  positions,  and  a 
curve  drawn  through  them  on  the  plane  of  N  would  be  the 
centrode  of  the  motion  of  M  relatively  to  N.  If,  now,  M  be 
fixed  and  N  moved  so  that  the  relative  motion  is  the  same  as 
before,  the  centrode  of  the  motion  of  N  relatively  to  M  may  be 
located  upon  the  plane  of  M.  Each  centrode  being  the  locus 
of  the  centres  in  its  plane,  the  two  centrodes  would  always  have 
one  point  in  common,  that  point  being  the  centre  of  the  relative 
motion  of  the  two  bodies  at  the  particular  instant.  (Otherwise 

*  Centrode  is  here  used  in  preference  to  "  centroid,"  proposed  by  Professor 
Kennedy,  because  the  latter  term  has  grown  to  be  generally  accepted  in  mathe- 
matics as  synonymous  with  "  center  of  mass." 


14  MACHINE  DESIGN. 

expressed,  the  corresponding  centrodes  are  two  curves  simul- 
taneously generated  in  the  two  planes  by  a  common  tracing 
point.)  Now,  since  the  centre  of  the  relative  motion  of  two 
bodies  is  a  point  at  which  they  have  no  relative  motion,  and 
since  the  points  of  the  centrodes  become  successively  the  cen- 
tres of  the  relative  motion,  it  follows  that  as  the  motion  goes 
on,  the  centrodes  would  roll  upon  each  other  without  slipping. 
Therefore,  if  the  centrodes  are  drawn,  and  rolled  upon  each 
other  without  slipping,  the  bodies  M  and  AT"  will  have  the  same 
relative  motion  as  before.  From  this  it  follows  that  the  rel- 
ative plane  motion  of  two  bodies  may  be  reproduced  by  rolling 
together,  without  slipping,  the  centrodes  of  that  motion. 

ii.  Pairs  of  Motion  Elements. — The  external  and  internal 
surfaces  by  which  motion  is  constrained,  as  in  Figs.  2  and  3,  may 
be  called  pairs  of  motion  elements.  The  pair  in  Fig.  2  is  called 
a  turning  pair,  and  the  pair  in  Fig.  3  is  called  a  sliding  pair. 

The  helical  surfaces  by  which  a  nut  and  screw  engage  with 
each  other  are  called  a  twisting  pair.  These  three  pairs  of 
motion  elements  have  their  surfaces  in  contact  throughout.  They 
are  called  lower  pairs.  Another  class,  called  higher  pairs,  have 
contact  only  along  elements  of  their  surfaces.  Examples. — Cams 
and  toothed  wheels. 


CHAPTER    II. 

MOTION  IN  MECHANISMS. 

12.  Linkages  or  Motion  Chains;   Mechanisms. 

In  Fig.  7,  b  is  joined  to  c  by  a  turning  pair; 
c          "          d     "   sliding     " 
d          "          a      "  turning    " 


FIG.  7. 

Evidently  there  is  complete  constrainment  of  the  relative 
motion  of  a,  b,  c,  and  d.  For,  d  being  fixed,  if  any  motion  occurs 
in  either  a,  b,  or  c,  the  other  two  must  have  a  predetermined 
corresponding  motion. 

c  may  represent  the  cross-head,  b  the  connecting-rod,  and  a 
the  crank  of  a  steam-engine  of  the  ordinary  type.  If  c  were 
rigidly  attached  to  a  piston  upon  which  the  expansive  force  of 
steam  acts  toward  the  right,  a  must  rotate  about  ad.  This 
represents  a  machine.  The  members  a,  6,  c,  and  d  may  be 
represented  for  the  study  of  relative  motions  by  the  diagram, 
Fig.  8. 

This  assemblage  of  bodies,  connected  so  that  there  is  complete 
constrainment  of  motion,  may  be  called  a  motion  chain  or  linkage, 

15 


i6 


MACHINE  DESIGN. 


and  the  connected  bodies  may  be  called  links.  The  chain 
shown  is  a  simple  chain,  because  no  link  is  joined  to  more 
than  two  others.  If  any  links  of  a  chain  are  joined  to  more 
than  two  others,  the  chain  is  a  compound  chain.  Examples  will 
be  given  later. 

When  one  link  of  a  chain  is  fixed,  i.e.,  when  it  becomes  the 
standard  to  which  the  motion  of  the  others  is  referred,  the  chain 
is  called  a  mechanism.  Fixing  different  links  of  a  chain  gives 
different  mechanisms.  Thus  in  Fig.  8,  if  d  is  fixed,  the  mechanism 
is  that  which  is  used  in  the  usual  type  of  steam-engine,  as  in 
Fig.  7.  It  is  called  the  slider-crank  mechanism. 

But  if  a  is  fixed,  the  result  is  an  entirely  different  mechanism ; 
for  b  would  then  rotate  about  the  permanent  center  ab,  d  would 
rotate  about  the  permanent  center  ad,  while  c  would  have  a  more 
complex  motion,  rotating  about  a  constantly  changing  centro, 
whose  path  may  be  found. 


Fixing  b  or  c  would  give,  in  each  case,  a  still  different  mechan- 


ism. 


13.  Location  of  Centres. — In  Fig.  8,  d  is  fixed  and  it  is  re- 
quired to  find  the  centers  of  rotation,  either  permanent  or  in- 


MOTION  IN  MECHANISMS.  17 

stantaneous,  of  the  other  three  links.  The  motion  of  a,  relative 
to  the  fixed  link  d,  is  rotation  about  ad,  the  axis  of  the  turning 
pair  joining  a  and  d.  When  two  links  are  joined  by  a  turning 
pair  their  centre  is  always  at  the  axis  of  the  pair.  When  two 
links  are  joined  by  a  sliding  pair  their  centre  lies  at  infinity  in 
a  direction  normal  to  their  relative  motion  of  translation.  The 
motion  of  c  relative  to  d  is  translation,  or  rotation  about  a  centre 
ed,  at  infinity  vertically.  The  centre  ab  lies  at  the  axis  of  the 
turning  pair  joining  a  and  b.  This  point  may  be  considered 
as  a  point  in  a  or  b;  in  either  case  it  can  have  but  one  direction 
of  instantaneous  motion  relative  to  any  one  standard.  As  a 
point  in  a  its  motion,  relative  to  d,  is  rotation  about  ad.  For 
the  instant,  then,  it  is  moving  along  a  tangent  to  the  circle 
through  ab.  But,  as  a  point  in  b,  its  direction  of  instantaneous 
motion  relative  to  d  must  be  the  same,  and  hence  its  motion 
must  be  rotation  about  some  point  in  the  line  ad-ab,  extended 
if  necessary.  Also,  b  has  a  point,  be,  in  common  with  c;  and 
by.  the  same  reasoning  as  above  be,  as  a  point  in  b,  rotates  for 
the  instant  about  some  point  of  the  vertical  line  through  be. 
Now  ab  and  be  are  points  of  a  rigid  body,  and  one  rotates  for 
the  instant  about  some  point  of  AB,  and  the  other  rotates  foi 
the  instant  about  some  point  of  CD;  hence  both  ab  and  be  (as 
well  as  all  other  points  of  b)  must  rotate  about  the  intersection 
of  AB  and  CD.  Hence  bd  is  the  centre  of  the  motion  of  b 
relative  to  d. 

The  motion  of  a  may  be  referred  to  c  (fixed),  and  ac  will  be 
found  (by  reasoning  like  that  applied  to  b)  to  lie  at  the  inter- 
section of  the  lines  EF  and  GH. 

The  motion  chain  in  Fig.  8,  as  before  stated,  is  called  the 
slider-crank  chain. 

14.  Centres  of  the  Relative  Motion  of  Three  Bodies  are  always 
in  the  Same  Straight  Line. — In  Fig.  8  it  will  be  seen  that  the 
three  centres  of  any  three  links  lie  in  the  same  straight  line. 


i8 


MACHINE  DESIGN. 


Thus  ad,  ab,  and  bd  are  the  centres  of  the  links  a,  b,  and  d.  This 
is  true  of  any  other  set  of  three  links. 

Proof. — Let  a,  b,  and  c,  Fig.  &A,  be  any  three  bodies  having 
relative  plane  motion.  Consider  a  fixed.  There  will  be  a 
centre,  ab,  of  the  relative  motion  of  b  and  a;  likewise  there  will 
be  a  centro,  ac,  of  the  relative  motion  of  c  and  a.  Let  their 
positions  be  assumed  as  represented.  There  will  also  be  a  centro 
be  and  it  must  lie  either  on  the  line  joining  ab  and  ac  or  off  that 
line.  Assume  it  to  lie  off  the  line,  as  shown. 


FIG.  8A. 


By  definition  it  is  a  point  of  both  b  and  c,  but  as  a  point  of 
either  it  must  have  the  same  instantaneous  motion  relative  to 
any  other  link,  such  as  a.  If  be  lie  off  the  line  ab-ac  as  shown, 
as  a  point  of  b  relatively  to  a  it  has  the  instantaneous  motion 
M,  normal  to  the  line  joining  it  to  ab.  As  a  point  of  c  relatively 
to  a  it  has  the  instantaneous  motion  N,  normal  to  the  line  joining 
it  to  ac.  But  by  the  definition  M  and  N,  being  the  instantaneous 
motion  of  the  same  point  relative  to  a,  must  coincide.  In  such 
case  the  normal  lines  bc-ab  and  bc-ac  must  coincide.  No  loca- 
tion of  bo  off  the  line  ab-ac  can,  therefore,  fulfill  the  conditions 
of  the  definition;  they  can  only  be  fulfilled  by  a  location  on  the 
line  ab-ac.  Hence  it  may  be  stated:  The  three  centros  of  any 
three  bodies  having  relative  plane  motion  must  lie  on  a  straight 


MOTION  IN   MECHANISMS.  1 9 

line.    This  important  proposition  is  called  Kennedy's  Theorem, 
after  its  discoverer. 

15.  Lever-crank  Chain.     Location  of  Centres. — Fig.  9  shows 
a  chain  of  four  links  of  unequal  length  joined  to  each  other  by 


FIG.  9. 

turning  pairs.  The  centres  ab,  ad,  cd,  and  be  may  be  located  at 
once,  since  they  are  at  the  centers  of  turning  pairs  which  join 
adjacent  links  to  each  other.  The  centres  of  the  relative  motion 
of  bj  c,  and  d  are  be,  cd,  and  bd;  and  these  must  be  in  the 
same  straight  line.  Hence  bd  is  in  the  line  B.  The  centres 
of  the  relative  motion  of  a,  b,  and  d  are  ab,  bd,  and  ad; 
and  these  also  must  lie  in  a  straight  line.  Hence  bd  is  in 
the  line  A.  Being  at  the  same  time  in  A  and  B,  it  must  be 
at  their  intersection.  By  employing  the  same  method  ac  may 
be  found. 

16.  The  Constrainment  of  Motion  in  a  linkage  is  inde- 
pendent of  the  size  of  the  motion  elements.  As  long  as  the 
cylindrical  surfaces  of  turning  pairs  have  their  axes  unchanged, 
the  surfaces  themselves  may  be  of  any  size  whatever,  and  the 
motion  is  unchanged.  The  same  is  true  of  sliding  and  twisting 
pairs. 

In  Fig.  10,  suppose  the  turning  pair  connecting  c  and  d  to  be 
enlarged  so  that  it  includes  be.  The  link  c  now  becomes  a 
cylinder,  turning  in  a  ring  attached  to,  and  forming  part  of, 


20  MACHINE  DESIGN. 

the  link  d.     be  becomes  a  pin  made  fast  in  c  and  engaging  with 
an  eye  at  the  end  of  b.     The  centres  are  the  same  as  before 


v\ 
FIG.  10. 

the  enlargement  of  the  pair  cd,  and  hence  the  relative  motion 
is  the  same. 

In  Fig.  ii  the  circular  portion  immediately  surrounding  cd 
is  attached  to  d.  The  link  c  now  becomes  a  ring  moving  in  a 
circular  slot.  This  may  be  simplified  as  in  Fig.  12,  whence  c 
becomes  a  curved  block  moving  in  a  limited  circular  slot  in  d. 
The  centres  remain  as  before,  the  relative  motion  is  the  same, 
and  the  linkage  is  essentially  unchanged. 

If,  in  the  slider-crank  mechanism,  the  turning  pair  whose 
axis  is  ab  be  enlarged  till  ad  is  included,  as  in  Fig.  13,  the  motion 
of  the  mechanism  is  unchanged,  but  the  link  a  is  now  called 
an  eccentric  instead  of  a  crank.  This  mechanism  is  usually 
used  to  communicate  motion  from  the  main  shaft  of  a  steam- 
engine  to  the  valve.  It  is  used  because  it  may  be  put  on  the 
main  shaft  anywhere  without  interfering  with  its  continuity  and 
strength. 

17.  Slotted  Cross-head. — The  mechanism  shown  in  Fig.  14 
is  called  the  "  slotted  cross-head  mechanism"  Its  centres  may 
be  found  from  principles  already  given. 


MOTION  IN  MECHANISMS. 


21 


This  mechanism  is  often  used  as  follows:  One  end  of  c,  as 
E,  is  attached  to  a  piston  working  in  a  cylinder  attached  to  d. 
This  piston  is  caused  to  reciprocate  by  the  expansive  force  of 
steam  or  some  other  fluid.  The  other  end  of  c  is  attached  to 


FIG.  ii. 


FIG.  12. 

another  piston,  which  also  works  in  a  cylinder  attached  to  d. 
This  piston  may  pump  water  or  compress  gas  (for  example 
small  ammonia  compressors  for  refrigerating  plants).  The 
crank  a  is  attached  to  a  shaft,  the  projection  of  whose  axis  is 
ad.  This  shaft  also  carries  a  fly-wheel  which  insures  approxi- 
mately uniform  rotation. 


22 


MACHINE  DESIGN. 


1 8.  Location  of  Centres  in  a  Compound  Mechanism. — It  is 

required  to  find  the  centres  of  the  compound  linkage,  Fig.  15. 
In  any  linkage,  each  link  has  a  centro  relatively  to  every  other 


FIG.  13. 

link;   hence,  if  the  number  of  links  =  n,  the  number  of  centres  = 
n(n  —  i).     But  the  centro  ab  is  the  same  as  ba\  i.e.,  each  centro 


FIG.  14. 

is  double.     Hence  the  number  of  centros  to  be  located  for  any 
linkage =—       — .     In  the  linkage  Fig.  15,  the  number  of  centros 
6X5 


*  The  links  are    a,  b,  c,  d,  e,  and  /. 
The  centros:     ab  be   cd  de  ef 
ac  bd  ce   df 
ad  be   cf 
ae  bf 
af 


MOTION  IN  MECHANISMS.  23 

The  portion  above  the  link  d  is  a  slider-crank  chain,  and 
the  character  of  its  motion  is  in  no  way  affected  by  the  attachment 
of  the  part  below  d.  On  the  other  hand,  the  lower  part  is  a 
lever-crank  chain,  and  the  character  of  its  motion  is  not  affected 
by  its  attachment  to  the  upper  part.  The  chain  may  therefore 
be  treated  in  two  parts,  and  the  centros  of  each  part  may  be 
located  from  what  has  preceded.  Each  part  will  have  six  centros, 
and  twelve  would  thus  be  located,  ad,  however,  is  common  to 


bd 


ed 


FIG.  15. 

the  two  parts,  and  hence  only  eleven  are  really  found.  Four 
centros,  therefore,  remain  to  be  located.  They  are  be,  c),  bf,  and 
ce.  To  locate  be,  consider  the  three  links  a,  b,  and  e,  and  it 
follows  that  be  is  in  the  line  A  passing  through  ab  and  ae;  con- 
sidering b,  d,  and  e,  it  follows  that  be  is  in  the  line  B  through  bd 
and  de.  Hence  be  is  at  the  intersection  of  A  and  B.  Similar 
methods  locate  the  other  centros. 

In  general,  for  rinding  the  centros  of  a  compound  linkage  of 


24  MACHINE  DESIGN. 

six  links,  consider  the  linkage  to  be  made  up  of  two  simple  chains; 
and  find  their  centros  independently  of  each  other.  Then  take 
the  two  links  whose  centro  is  required,  together  with  one  of 
the  links  carrying  three  motion  elements  (as  a,  Fig.  15).  The 
centros  of  these  links  locate  a  straight  line,  A,  which  contains 
the  required  centro.  Then  take  the  two  links  whose  centro  is 
required,  together  with  the  other  link  which  carries  three  motion 
elements.  A  straight  line,  B,  is  thereby  located,  which  contains 
the  required  centro,  and  the  latter  is  therefore  at  the  intersection 
of  A  and  B. 

19.  Velocity  is  the  rate  of  motion,  or  motion  per  unit  time. 

Linear  velocity  is  linear  space  moved  through  in  unit  time; 
it  may  be  expressed  in  any  units  of  length  and  time;  as,  miles 
per  hour,  feet  per  minute  or  per  second,  etc. 

Angular  velocity  is  angular  space  moved  through  in  unit  time. 
In  machines,  angular  velocity  is  usually  expressed  in  revolutions 
per  minute  or  per  second. 

The  linear  space  described  by  a  point  in  a  rotating  body,  or 
its  linear  velocity,  is  directly  proportional  to  its  radius,  or  its 
distance  from  the  axis  of  rotation.  This  is  true  because  arcs 
are  proportional  to  radii. 

If  A  and  B  are  two  points  in  a  rotating  body,  and  if  r\  and  r^ 
are  their  radii,  then  the  ratio  of  linear  velocities 

linear  veloc.  A     r\ 
linear  veloc.  B     r2' 

This  is  true  whether  the  rotation  is  about  a  center  or  a  centro ; 
i.e.,  it  is  true  either  for  continuous  or  instantaneous  rotation. 
Hence  it  applies  to  all  cases  of  plane  motion  in  machines;  because 
all  plane  motion  in  machines  is  equivalent  to  either  continuous 
or  instantaneous  rotation  about  some  point. 

To  find  the  relation  of  linear  velocity  of  two  points  in  a  machine 
member,  therefore,  it  is  only  necessary  to  find  the  relation  of 


MOTION  IN  MECHANISMS.  25 

the  radii  of  the  points.  The  latter  relation  can  easily  be  found 
when  the  center  or  centro  is  located. 

20.  A  vector  quantity  possesses  magnitude  and  direction.  It 
may  be  represented  by  a  straight  line,  because  the  latter  has 
magnitude  (its  length)  and  direction.  Thus  the  length  of  a 
straight  line,  AB,  may  represent,  upon  some  scale,  the  magnitude 
of  some  vector  quantity,  and  it  may  represent  the  vector  quantity's 
direction  by  being  parallel  to  it,  or  by  being  perpendicular  to  it. 
For  convenience  the  latter  plan  will  here  be  used.  The  vector 
quantities  to  be  represented  are  the  linear  velocities  of  points 
in  mechanisms.  The  lines  which  represent  vector  quantities  are 
called  vectors. 

A  line  which  represents  the  linear  velocity  of  a  point  will 
be  called  the  linear  velocity  vector  of  the  point.  The  symbol  of 
linear  velocity  will  be  VI.  Thus  VIA  is  the  linear  velocity  of 
the  point  A.  Also  Va  will  be  used  as  the  symbol  of  angular 
velocity.  t 

If  the  linear  velocity  and  radius  of  a  point  are  known,  the 
angular  velocity,  or  the  number  of  revolutions  per  unit  time, 
may  be  found;  since  the  linear  velocity  H-  length  of  the  circum- 
ference in  which  the  point  travels  =  angular  velocity. 

All  points  of  a  rigid  body  have  the  same  angular  velocity. 

If  the  radii,  and  ratio  of  linear  velocities  of  two  points,  in 
different  machine  members  are  known,  the  ratio  of  the  angular 
velocities  of  the  members  may  be  found  as  follows: 

Let  A  be  a  point  in  a  member  M,  and  B  a  point  in  a  member 
N.  r\  =  radius  of  A  ;  r^  =  radius  of  B.  VIA  and  VIB  represent 

VIA 
the  linear  velocities  of  A    and  B,   whose   ratio,  T/TDJ  is  known. 

Then  VaA=~    and     VaB=~  . 

27ZT2 


_.  _  r2     VaM 

VaB  ~  27rn  X  VIB  ~  VIB  X  n  ~  VaN  ' 


26  MACHINE  DESIGN. 

If  M  and  N  rotate  uniformly  about  fixed  centers,  the  ratio 

VaM 

is  constant.     If  either  M  or  N  rotates  about  a  centre,  the 


ratio  is  a  varying  one. 

21.  To  find  the  relation  of  linear  velocity  of  two  points  in 
the  same  link,  it  is  only  necessary  to  measure  the  radii  of  the 
points,  and  the  ratio  of  these  radii  is  the  ratio  of  the  linear  veloci- 
ties of  the  points. 

In  Fig.  1  6,  let  the  smaller  circle  represent  the  path  of  A, 
the  center  of  the  crank-pin  of  a  slider-crank  mechanism;  the 
link  d  being  fixed.  Let  the  larger  circle  represent  the  rim  of  a 
pulley  which  is  keyed  to  the  same  shaft  as  the  crank.  The 
pulley  and  the  crank  are  then  parts  of  the  same  link.  The  ratio 

VIA 
of  velocity  of  the  crank-pin  center  and  the  pulley  surf  ace 


=—  .     In  this  case  the  link  rotates  about  a  fixed  center.     The 
r\ 

same  relation  holds,   however,   when  the  link  rotates  about  a 
centre. 


FIG.  1 6. 

22.  Velocity  Diagram  of  Slider-crank  Chain. — In    Fig.    17, 

Vlab     ab-bd 
the  link  d  is  fixed  and  T77I"  =  A  _M  •      BY  similar  triangles  this 

expression  is  also  equal  to    ~.  .     Hence,  if  the  radius  of  the 


MOTION  IN  MECHANISMS. 


27 


crank  circle  be  taken  as  the  -vector  of  the  constant  linear  velocity 
of  ab,  the  distance  cut  off  on  the  vertical  through  O  by  the  line  oj 
the  connecting-rod  (extended  ij  necessary)  will  be  the  vector  o]  the 
linear  velocity  of  be.  Project  A  horizontally  upon  bc-bd,  locating 
B.  Then  bc-B  is  the  vector  of  VI  of  the  slider,  and  may  be 

FIG.  17. 


FIG.  18. 

used  as  an  ordinate  of  the  linear  velocity  diagram  of  the  slider. 
By  repeating  the  above  construction  for  a  series  of  positions, 
the  ordinates  representing  the  VI  of  be  for  different  positions  of 
the  slider  may  be  found.  A  smooth  curve  through  the  extremi- 
ties of  these  ordinates  is  the  velocity  curve,  from  which  the  Vis 


28  MACHINE  DESIGN. 

of  all  points  of  the  slider's  stroke  may  be  read.  The  scale  of 
velocities,  or  the  linear  velocity  represented  by  one  inch  of  ordi- 
nate,  equals  the  constant  linear  velocity  of  ab  divided  by  O-ab 
in  inches. 

23.  Velocity  Diagram  of  Lever-crank  Chain. — It  is  required 
to  find  VI  of  be  during  a  cycle  of  action  of  the  mechanism  shown 
in  Fig.  18,  d  being  fixed,  and  VI  of  ab  being  constant.  The 
two  points  ab  and  be  may  both  be  considered  in  the  link  b. 
All  points  in  b  move  about  bd  relatively  to  the  fixed  link. 

Vlab     ab-bd 
Hence 


Vlbc      bc-bd  ' 

For  most  positions  of  the  mechanism  bd  will  be  so  located  as  to 
make  it  practically  impossible  to  measure  these  radii,  but  a  line, 
as  MNj  drawn  parallel  to  b  cuts  off  on  the  radii  portions  which 
are  proportional  to  the  radii  themselves,  and  hence  proportional 
to  the  Vis  of  the  points.  Hence 

Vlab 


Vlbc  ~  bc-N  * 

The  arc  in  which  be  moves  may  be  divided  into  any  number  of 
parts,  and  the  corresponding  positions  of  ab  may  be  located.  A 
circle  through  M,  with  ad  as  center,  may  be  drawn,  and  the 
constant  radial  distance  ab-M  may  represent  the  constant 
velocity  of  ab.  Through  Mi,  MZ,  etc.,  draw  lines  parallel  to  the 
corresponding  positions  of  b,  and  these  lines  will  cut  off  on  the 
corresponding  line  of  c  a  distance  which  represents  VI  of  be. 
Through  the  points  thus  determined  the  velocity  diagram  may 
be  drawn,  and  the  VI  of  be  for  a  complete  cycle  is  determined. 
The  scale  of  velocities  is  found  as  in  Sec.  22. 

24.  The  relation  of  linear  velocity  of  points  not  in  the  same 
link  may  also  be  found. 


MOTION  IN  MECHANISMS. 


29 


VI  of  A 


Required  —, — -r~B  referred   to   d  as   the   fixed  link,   Fig.    19. 

The  centro  ab  is  a  point  in  common  to  a  and  b,  the  two  links 
considered.  Consider  ab  as  a  point  in  a;  and  its  VI  is  to  that 
of  A  as  their  radii  or  distances  from  ad.  Draw  a  vector  triangle 
with  its  sides  parallel  to  the  triangle  formed  by  joining  A,  ab, 


FIG.  19. 

and  ad.  Then  if  the  side  A\  represent  the  VI  of  A,  the  side  a\b\ 
will  represent  the  VI  of  ab.  Consider  ab  as  a  point  in  &,  and 
its  VI  is  to  that  of  B  as  their  radii,  or  distances  to  bd.  Upon 
the  vector  a\b\  draw  a  triangle  whose  sides  are  parallel  to  those 
of  a  triangle  formed  by  joining  ab,  bd,  and  B.  Then,  from 
similar  trianrles,  the  side  BI  is  the  vector  of  $'s  linear  velocity. 


Hence 


VI  of  A     vector  A 


VI  of  B    vector 


The  path  of  B  during  a  complete  cycle  may  be  traced,  and  the 
VI  for  a  series  of  points  may  be  found,  by  the  above  method;  then 
the  vectors  may  be  laid  off  on  normals  to  the  path  through  the 
points;  the  velocity  curve  may  be  drawn;  and  the  velocity  of 
B  at  all  points  becomes  known. 


MACHINE    DESIGN. 


In  general,  to  find  the  instantaneous  motion,  in  direction  and 
velocity,  of  a  point  X  in  any  link  x,  relatively  to  a  fixed  link  d, 
given  the  motion  of  any  other  link  a  relatively  to  d:  First,  locate 
the  centre  ad.  All  points  of  a  rotate  about  this  centro  relatively 
to  d  and  have  linear  velocities  proportional  to  their  distances 
from  ad.  Second,  locate  centro  ax.  Its  velocity  relatively  to 
d  is  known,  likewise  its  direction  of  motion,  since  it  is  a  point 
of  a  and  therefore  rotates  about  ad  with  a  linear  velocity  pro- 
portional to  its  distance  from  ad.  As  a  point  of  x  it  has  this 
same  motion  relatively  to  d.  Third,  locate  dx.  All  points  of 
x  relatively  to  d  rotate  about  this  centro  and  have  linear  velocities 
proportional  to  their  distances  from  dx.  Hence  the  linear  veloc- 
ity of  the  point  X  is  to  the  known  linear  velocity  of  the  point 

X-dx 

ax  as  -    — — . 
ax—doc 

25.  Angularity  of  Connecting-rod. — The  diagram  of  VI  of 
the  slider-crank  mechanism,  Fig.  17,  is  unsymmetrical  with 
respect  to  a  vertical  axis  through  its  center.  This  is  due  to  the 
angularity  of  the  connecting-rod,  and  may  be  explained  as  follows : 
In  Fig.  20,  AO  is  one  angular  position  of  the  crank,  and  BO 
is  the  corresponding  angular  position  on  the  other  side  of  the 

vertical  through  the 
center  of  rotation. 
The  corresponding 
positions  of  the  slider 
are  as  shown.  But 
for  position  A  the 
line  of  the  connecting- 
rod,  C,  cuts  off  on 
the  vertical  through 

O  a  vector  Oa,  which  represents  the  slider's  velocity.  For  posi- 
tion B  the  vector  of  the  slider's  velocity  is  Ob  and  the  velocity 
diagram  is  unsymmetrical. 


FIG.  20. 


MOTION  IN  MECHANISMS.  31 

If  the  connecting-rod  were  parallel  to  the  direction  of  the 
slider's  motion  in  all  positiors,  as  in  the  slotted  cross  head 
mechanism  (see  Fig.  14),  the  vector  cut  off  on  the  vertical  through 
O  would  be  the  same  for  position  A  and  position  B  and  the 
velocity  diagram  would  be  symmetrical. 

Since  the  velocity  diagram  is  symmetrical  with  a  parallel 
connecting-rod  and  unsymmetrical  with  an  angular  connecting- 
rod,  with  all  other  conditions  constant,  it  follows  that  the  lack 
of  symmetry  is  due  to  the  angularity  of  the  connecting-rod. 

The  velocity,  diagram  for  the  slotted  cross-head  mechanism 
is  symmetrical  with  respect  to  both  vertical  and  horizontal  axes 
through  its  center.  In  fact,  if  the  crank  radius  (  =  length  of 
link  a)  be  taken  as  the  vector  of  the  VI  of  a&,  the  linear  velocity 
diagram  of  the  slider  becomes  a  circle  whose  radius  =the  length  of 
the  link  a.  Hence  the  crank  circle  itself  serves  for  the  linear  velocity 
diagram,  the  horizontal  diameter  representing  the  path  of  the  slider. 

26.  Angularity  of  Connecting-rod,  Continued. — During  a  por- 
tion of  the  cycle  of  the  slider-crank  mechanism,  the  slider's  VI 
is    greater   than    that    of   ab. 
This  is  also  due  to  the  an- 
gularity of  the  connecting-rod, 
and  may  be  explained  as  fol- 
lows: In  Fig.  21,  as  the  crank 
moves   up   from   the   position 
x,  it  will  reach  such  a  position, 

A,  that  the  line  of  the  connecting-rod  extended  will  pass 
through  B.  OB  in  this  position  is  the  vector  of  the  linear  veloc- 
ity of  both  ab  and  the  slider,  and  hence  their  linear  velocities 
are  equal.  When  ab  reaches  B,  the  line  of  the  connecting-rod 
passes  through  B ;  and  again  the  vectors — and  hence  the  linear 
velocities — of  ab  and  the  slider  are  equal.  For  all  positions 
between  A  and  B  the  line  of  the  connecting-rod  will  cut  OB 
outside  of  the  crank  circle;  and  hence  the  linear  velocity  of  the 


32  MACHINE  DESIGN. 

slider  will  be  greater  than  that  of  ab.  This  result  is  due  to  the 
angularity  of  the  connecting-rod,  because  if  the  latter  remained 
always  horizontal,  its  line  could  never  cut  OB  outside  the  circle. 
It  follows  that  in  the  slotted  cross-head  mechanism  the  maximum 
VI  of  the  slider  =  the  constant  VI  of  ab.  The  angular  space  BOA, 
Fig.  21,  throughout  which  VI  of  the  slider  is  greater  than  the  VI 
of  ab,  increases  with  increase  of  angularity  of  the  connecting-rod  • 
i.e.,  it  increases  with  the  ratio 

Length  of  crank 
Length  of  connecting-rod  " 

27.  Quick-return  Mechanisms. — A    slider    in    a    mechanism 
often  carries  a  cutting-tool,  which  cuts  during  its  motion  in  one 
direction,  and  is  idle  during  the  return  stroke.     Sometimes  the 
slider  carries  the  piece  to  be  cut,  and  the  cutting  occurs  while 
it  passes  under  a  tool  made  fast  to  the  fixed  link,  the  return 
stroke  being  idle. 

The  velocity  of  cutting  is  limited.  If  the  limiting  velocity 
be  exceeded,  the  tool  becomes  so  hot  that  it  becomes  unfit  for 
cutting.  The  limit  of  cutting  velocity  depends  on  the  nature 
of  the  material  to  be  cut,  and  the  quality  of  the  tool-steel  used. 
There  is  no  limit  of  this  kind,  however,  to  the  velocity  during 
the  idle  stroke ;  and  it  is  desirable  to  make  it  as  great  as  possible, 
in  order  to  increase  the  product  of  the  machine.  This  leads 
to  the  design  and  use  of  "  quick-return"  mechanisms. 

28.  Slider-crank    Quick    Return.  —  If,    in    a    slider-crank 
mechanism,  the  center  of    rotation  of    the  crank  be  moved,  so 
that  the  line  of  the  slider's  motion  does  not  pass  through  it,  the 
slider  will  have  a  quick-return  motion. 

In  Fig.  22,  when  the  slider  is  in  its  extreme  position  at  the 
right,  A,  the  crank-pin  center  is  at  D.  When  the  slider  is  at  B, 
the  crank-pin  center  is  at  C.  If  rotation  is  as  indicated  by  the 
arrow,  then,  while  the  slider  moves  from  B  to  A,  the  crank-pin 


MOTION  IN  MECHANISMS.  33 

center  moves  from  C  over  to  D.  And  while  the  slider  returns  from 
A  to  B,  the  crank-pin  center  moves  under  from  D  to  C.  If  the 
VI  of  the  crank-pin  center  be  as- 
sumed constant,  the  time  occu- 
pied  in  moving  from  D  to  C  is' 
less  than  that  from  C  to  D. 
Hence  the  time  occupied  by  the 
slider  in  moving  from  B  to  A  is 
greater  than  that  occupied  in 
moving  from  A  to  B.  The  mean 
velocity  during  the  forward 

stroke  is  therefore  less  than  during  the  return  stroke.  Or  the 
slider  has  a  "  quick-return  "  motion. 

It  is  required  to  design  a  mechanism  of  this  kind  for  a  length 
of  stroke  =BA  and  for  a  ratio 

mean  VI  forward  stroke     5 

mean  VI  return  stroke      7  * 

i 

The  mean  velocity  of  either  stroke  is  inversely  proportional  to 
the  time  occupied,  and  the  time  is  proportional  to  the  correspond- 
ing angle  described  by  the  crank.  Hence 

mean  velocity  forward     5      angle  /? 
mean  velocity  return      7      angle  a 

It  is  therefore  necessary  to  divide  360°  into  two  parts  which 
are  to  each  other  as  5  to  7.  Hence  a  =  210°  and  /?  =  i5o°.  Ob- 
viously 0  =  i8o°— /?  =  30°.  Place  the  30°  angle  of  a  drawing 
triangle  so  that  its  sides  pass  through  B  and  A.  This  condition 
may  be  fulfilled  and  yet  the  vertex  of  the  triangle  may  occupy 
an  indefinite  number  of  positions.  By  trial  O  may  be  located  so 
that  the  crank  shall  not  interfere  with  the  line  of  the  slider.* 


*To  avoid  cramping  of  the  mechanism,  the  angle  BAD  should  equal  or  excee-i 

i35°- 


34  MACHINE  DESIGN. 

O  being  located  tentatively,  it  is  necessary  to  find  the  correspond- 
ing lengths  of  crank  a  and  connecting-rod  b.  When  the  crank- 
pin  center  is  at  D,  AO  =  b  -a;  when  it  is  at  C,  BO  =  b  +a.  AO 
and  BO  are  measurable  values  of  length  whose  difference  =  za  ; 
hence  a  and  b  may  be  found,  the  crank  circle  may  be  drawn, 
and  the  velocity  diagrams  may  be  constructed  as  in  Fig.  17; 
remembering  that  the  distance  cut  off  upon  a  vertical  through  O, 
by  the  line  of  the  connecting-rod,  is  the  vector  of  the  VI  of  the 
slider  for  the  corresponding  position  when  the  VI  of  the  crank- 
pin  center  is  represented  by  the  crank  radius. 

It  is  required  to  make  the  maximum  velocity  of  the  forward 
stroke  of  the  slider  =  20  feet  per  minute,  and  to  find  the  corre- 
sponding number  of  revolutions  per  minute  of  the  crank.  The 
maximum  linear  velocity  vector  of  the  forward  stroke  =  the 
maximum  height  of  the  upper  part  of  the  velocity  diagram; 
call  it  Vl\.  Call  the  linear  velocity  vector  of  the  crank-pin  center 
F/2  =  crank  radius.  Let  x=  linear  velocity  of  the  crank-pin 
center.  Then 

F/i     20  ft.  per  minute 


20  ft.  per  minute  X  Viz 

- 


x  is  therefore  expressed  in  known  terms.  If  now  x,  the  space 
the  crank-pin  center  is  required  to  move  through  per  minute, 
be  divided  by  the  space  moved  through  per  revolution,  the  result 
will  equal  the  number  of  revolutions  per  minute  =JV; 


27rX  length  of  crank* 

29.  Lever-crank  Quick  Return  — Fig.  23  shows  a  compound 
mechanism.  The  link  d  is  the  supporting  frame  or  fixed  link, 
and  a  rotates  about  ad  in  the  direction  indicated,  communicating 


MOTION  IN  MECHANISMS. 


35 


motion  to  c  through  the  slider  b  so  that  c  vibrates  about  cd.  The 
link  e,  connected  to  c  by  a  turning  pair  at  ce,  causes  /  to  slide 
horizontally  on  another  part  of  the  frame  or  fixed  link  d.  The 
center  of  the  crank-pin,  ab,  is  given  a  constant  linear  velocity, 
and  the  slider,  /,  has  motion  toward  the  left  with  a  certain  mean 
velocity,  and  returns  toward  the  right  with  a  greater  mean  velocity^ 
This  is  true  because  the  slider  /  moves  toward  the  left  while  a 
moves  through  the  angle  a ;  and  toward  the  right  while  a  moves 
through  the  angle  /?.  But  the  motion  of  a  is  uniform,  and  hence 
the  angular  movement  a.  represents  more  time  than  the  angular 
movement  /?;  and  /,  therefore,  has  more  time  to  move  toward 
the  left  than  it  has  to  move  through  the  same  space  toward  the 
right.  It  therefore  has  a  "quick-return"  motion. 


df  at  vertical 
infinity 


FlG.  23. 

The  machine  is  driven  so  that  the  crank-pin  center  moves 
uniformly,  and  the  velocity,  at  all  points  of  its  stroke,  of  the 
slider  carrying  a  cutting-tool,  is  required.  The  problem,  there- 
fore, is  to  find  the  relation  of  linear  velocities  of  ef  and  ab  for  a 
series  of  positions  during  the  cycle;  and  to  draw  the  diagram 
of  velocity  of  ef. 

Solution. — ab  has  a  constant  known  linear  velocity.  The 
point  in  the  link  c  which  coincides,  for  the  instant,  with  ab,  re- 
ceives motion  from  ab,  but  the  direction  of  its  motion  is  different 


36  MACHINE  DESIGN. 

from  that  of  ab,  because  ab  rotates  about  ad,  while  the  coin- 
ciding point  of  c  rotates  about  cd.  If  ab-A  be  laid  off  repre- 
senting the  linear  velocity  of  ab,  then  ab-B  will  represent  the 
linear  velocity  of  the  coinciding  point  of  the  link  c.  Let  the 
latter  point  be  called  x. 

Locate  cf,  at  the  intersection  of  ef-ce  with  the  line  cd-df.  Now 
cj  and  x  are  both  points  in  the  link  c,  and  hence  their  linear 
velocities,  relatively  to  the  fixed  link  d,  are  proportional  to  their 
distances  from  cd.  These  two  distances  may  be  measured 
directly,  and  with  the  known  value  of  linear  velocity  of  x  =  ab-B 
give  three  known  values  of  a  simple  proportion,  from  which  the 
fourth  term,  the  linear  velocity  of  cf,  may  be  found. 

Or,  if  the  line  BD  be  drawn  parallel  to  cd-cf,  the  triangle 
B-D-ab  is  similar  to  the  triangle  cd-cj-ab,  and  from  the  simi- 
larity of  these  triangles  it  follows  that  BD  represents  the  linear 
velocity  of  cj  on  the  same  scale  that  ab-B  represents  the  linear 
velocity  of  x.  Hence  the  linear  velocity  of  c},  for  the  assumed 
position  of  the  mechanism,  becomes  known.  But  since  cj  is  a 
point  of  the  slider,  all  of  whose  points  have  the  same  linear  velocity 
because  its  motion  relatively  to  d  is  rectilinear  translation,  it 
follows  that  the  linear  velocity  of  cj  is  the  required  linear  velocity 
of  the  slider.  At  ef  erect  a  line  perpendicular  to  the  direction 
of  motion  of  the  slider  having  a  length  equal  to  BD. 

This  solution  may  be  made  for  as  many  positions  of  the 
mechanism  as  are  necessary  to  locate  accurately  the  velocity 
curve.  The  ordinates  of  this  curve  will,  of  course,  be  the  veloci- 
ties of  the  slider,  and  the  abscissas  the  corresponding  positions 
of  the  slider. 

Having  drawn  the  velocity  diagram,  suppose  that  it  is  required 
to  make  the  maximum  linear  velocity  of  the  slider  on  the  slow 
stroke  =Q  feet  per  minute.  Then  the  linear  velocity  of  the 
crank-pin  center  ab=y  can  be  determined  from  the  propor- 
tion 


MOTION  IN  MECHANISMS,  37 

y  vector  A  -ab 

Q  "maximum  ordinate  of  velocity  diagram' 

vector  A-ab 
' '  y~  ^maximum  ordinate  of  velocity  diagram* 

y 

If  r  =  the  crank  radius,  the  number  of  revolutions  per  minute  =  —  . 

When  this  mechanism  is  embodied  in  a  machine,  a  becomes 
a  crank  attached  to  a  shaft  whose  axis  is  at  ad.  The  shaft  turns 
in  bearings  provided  in  the  machine  frame.  The  crank  carries  a 
pin  whose  axis  is  at  a&,  and  this  pin  turns  in  a  bearing  in  the 
sliding  block  b.  The  link  c  becomes  a  lever  keyed  to  a  shaft 
whose  axis  is  at  cd.  This  lever  has  a  long  slot  in  which  the  block 
b  slides.  The  link  e  becomes  a  connecting-rod,  connected  to  both 
c  and  /  by  pin  and  bearing.  The  link  /  becomes  the  "cutter- 
bar"  or  "ram"  of  a  shaper:  the  part  which  carries  the  cutting- 
tool.  The  link  d  becomes  the  frame  of  the  machine,  which  not 
only  affords  support  to  the  shafts  at  ad  and  cd,  and  the  guiding 
surfaces  for  /,  but  also  is  so  designed  as  to  afford  means  for  holding 
the  pieces  to  be  planed,  and  supports  the  feed  mechanism. 


FIG.  24. 


30.  Whitworth  Quick  Return. — Fig.  24  shows  another  com- 
pound linkage,     d  is  fixed,  and  c  rotates  uniformly  about  cd, 


3  MACHINE  DESIGN. 

communicating  an  irregular  rotary  motion  to  a  through  the  slider 
b.  a  is  extended  past  ad  (the  part  extended  being  in  another 
parallel  plane),  and  moves  a  slider  /  through  the  medium  of 
a  link  e.  This  is  called  the  "  Whitworth  quick-return  mechanism." 
The  point  be,  at  which  c  communicates  motion  to  a,  moves  along 
a,  and  hence  the  radius  (measured  from  ad)  of  the  point  at  which 
a  receives  a  constant  linear  velocity  varies,  and  the  angular 
velocity  of  a  must  vary  inversely.  Hence  the  angular  velocity 
of  a  is  a  maximum  when  the  radius  is  a  minimum,  i.e.,  when 
a  and  c  are  vertical  downward;  and  the  angular  velocity  of  a 
is  minimum  when  the  radius  is  a  maximum,  i.e.,  when  a  and  c 
are  vertical  upward. 

31.  Problem. — To  design  a  Whitworth  Quick  Return  for  a 

given  ratio, 

mean  VI  of  /  forward 
mean  VI  of  /  returning* 

When  the  center  of  the  crank-pin,  C,  reaches  A,  the  point  D  will 
coincide  with  B,  the  link  c  will  occupy  the  angular  position  cd-B, 
and  the  slider  /  will  be  at  its  extreme  position  toward  the  left. 

When  the  point  C  reaches  F,  the  point  D  will  coincide  with 
E,  the  link  c  will  occupy  the  angular  position  cd-E,  and  the 
slider  /  will  be  at  its  extreme  position  toward  the  right. 

Obviously,  while  the  link  c  moves  over  from  the  position 
cd-E  to  the  position  cd-B,  the  slider  /  will  complete  its  forward 
stroke,  i.e.,  from  right  to  left.  While  c  moves  under  from  cd-B 
to  cd-E,  }  will  complete  the  return  stroke,  i.e.,  from  left  to  right. 
The  link  c  moves  with  a  uniform  angular  velocity,  and  hence  the 
mean  velocity  of  /  forward  is  inversely  proportional  to  the  angle 
/?  (because  the  time  consumed  for  the  stroke  is  proportional  to 
the  angle  moved  through  by  the  crank  c},  and  the  mean  velocity 
of  /  returning  is  inversely  proportional  to  a.  Or 

mean  VI  of  /  forward      a 
mean  VI  of  /  returning    @' 


MOTION  IN  MECHANISMS.  39 

For  the  design  the  distance  cd-ad  must  be  known.     This  may 
usually  be  decided  on  from  the  limiting  sizes  of  the  journals  at  cd 

and  ad.     Suppose  that  the  above  ratio  =-  =  -,  that  cd-ad  =  3", 

and  that  the  maximum  length  of  stroke  of  /=i2".  Locate  cd 
and  measure  off  vertically  downward  a  distance  equal  to  3", 
thus  locating  ad.  Draw  a  horizontal  line  through  ad.  The 
point  ej  of  the  slider  /  will  move  along  this  line.  Since 


and 


/.     a  =  150°   and        /?  =  2io°. 

Lay  off  a  from  cd  as  a  center,  so  that  the  vertical  line  through 
cd  bisects  it.  Draw  a  circle  through  B  with  cd  as  a  center,  B 
being  the  point  of  intersection  of  the  bounding  line  of  a  with  a 
horizontal  through  ad.  The  length  of  the  link  c  =  cd-B. 

The  radius  ad-C  must  equal  the  travel  of  /-f-2  =  6".  This 
radius  is  made  adjustable,  so  that  the  length  of  stroke  may  be 
varied.  The  connecting-rod,  e,  may  be  made  of  any  convenient 
length. 

32.  Problem.  —  To  draw  the  velocity  diagram  of  the  slider 
/  of  the  Whitworth  Quick  Return.  The  point  be,  Fig.  25,  as  a 
point  of  c  has  a  known  constant  linear  velocity  relative  to  d,  and 
its  direction  of  motion  is  always  at  right  angles  to  a  line  joining 
it  to  cd.  That  point  of  the  link  a  which  coincides  in  this  posi- 
tion of  the  mechanism  with  be,  receives  motion  from  be,  but  its 
direction  of  motion  relative  to  d  is  at  right  angles  to  the  line  be- 
ad. If  be-  A  represents  the  linear  velocity  of  be,  its  projection 
upon  bc-ad  extended  will  represent  the  linear  velocity  of  the 
point  of  a  which  coincides  with  be.  Call  this  point  x.  Locate 
the  centre  a/,  draw  the  line  aj-bc  and  extend  it  to  meet  the  vertical 
dropped  from  B  to  C.  The  centro  af  may  be  considered  as  a 


MACHINE  DESIGN. 


point  in  a,  and  its  linear  velocity  relative  to  d,  when  so  considered, 
is  proportional  to  its  distance  from  ad.     Hence 


VI  of  af 
VI  of  x 


ad— a] 
ad-bc  ' 


be  and  a: 


FIG.  25. 
But  the  triangles  ad-af-bc  and  B-C-bc  are  similar.     Hence 

VI  of  af     EC 
Vlofx~  B-bc 

This  means  that  EC  represents  the  linear  velocity  of  af  upon  the 
same  scale  that  B-bc  represents  the  linear  velocity  of  x.  But 
af  is  a  point  in  /,  and  all  points  in  /  have  the  same  linear  velocity 
relative  to  d  since  the  motion  is  rectilinear  translation;  hence 
EC  represents  the  linear  velocity  of  the  slider  /  for  the  given 
position  of  the  mechanism,  and  it  may  be  laid  off  as  an  ordinate 
of  the  velocity  curve.  This  solution  may  be  made  for  as  many 
positions  as  are  required  to  locate  accurately  the  entire  velocity 
curve  for  a  cycle  of  the  mechanism. 


CHAPTER  III. 

PARALLEL  OR  STRAIGHT-LINE  MOTIONS. 

33.  Watt  Parallel  Motion. — Rectilinear  motion  in  machines 
is  usually  obtained  by  means  of  prismatic  guides.     It  is  some- 
times  necessary,    however,    to    accomplish  the   same    result   by 
linkages. 

The  simplest  and  most  widely  known  linkage  used  for  giving 
rectilinear  motion  to  a  point  without  the  use  of  any  sliding  pairs 
is  the  so-called  Watt  Parallel  Motion.  It  is  one  of  the  numerous 
inventions  of  James  Watt  and  x 

consists    of    four     links,   three " £' 

moving  and  one  fixed,  all  con- 
nected by  turning  pairs,  d, 
Fig.  26,  is  the  fixed  link,  a 

\ 

rotates  relative  to  d  about  ad', 

FIG.  26. 
c  rotates  relative  to  d  about  cd. 

The  mechanism  is  shown  in  the  position  corresponding  to  the 
middle  of  its  motion.  Links  a  and  c  being  equal,  P  is  mid-point 
of  b. 

As  the  points  ab  and  be  swing  in  the  dotted  arcs,  the  point  P 
will  travel  in  approximately  a  straight  line.  The  whole  path  of 
P  is  a  lemniscate,  but  the  part  which  is  ordinarily  used  approaches 
very  closely  to  a  straight  line. 

34.  Parallelogram. — A  true  parallel  motion  is  given  by  the 
Parallelogram  which  is  shown  in  Fig.  27.     The  links  a,  b,  c,  and  e 
are  connected  to  each  other  by  turning  pairs,  and  the  linkage  is 
attached  to  the  fixed  link  d  by  a  turning  pair  at  ac.     (This  point 
is  also  ad  and  cd.) 

41 


MACHINE  DESIGN. 


The  lengths  ac  —  ab  and  ce  —  be  are  equal,  as  are  also  ac  —  ce  and 
ab  —  be.  The  point  P  is  fixed  on  e.  Draw  a  line  from  P  to  ac ;  it 
cuts  the  link  b  at  P'. 

By  similar  triangles, 

P-be      P-be 


ac  —  ce 


P-ce 
P-be 


.*.  P'  —  be  =ac  —  ce  (  ]  =  a    constant. 

\P-ceJ 

Therefore  the  point  P'  will  lie  at  the  same  position  on  b  for  all 
positions  of  the  mechanism.     Likewise,  by  similar  triangles,  the 

ratio  -  — —   =  a   constant    for    all    positions    of  the 

P'  —  ac       be—ce 

mechanism.     Since  the  line  P— P'  swings,  relative  to  d,  about  the 


d 

FIG.  27.  FIG.  28. 

pole  ad  (ac,  cd)  at  every  instant,  it  is  obvious  that  the  motions  of 
P  and  P'  relative  to  d  will  be  similar  to  each  other  in  every  respect 

P  -ac 


and  always  in  the  ratio 


It  follows  that,  if  either  of 


P'-ac 

these  points  is  guided  to  move  in  a  straight-line  path,  the  other 
point  is  constrained  to  move  in  a  similar  parallel  path.* 

*  The  following  demonstration  is  given  for  those  who  prefer  an  accurate 
proof : 

The  position  of  the  mechanism  in  Fig.  27  is  taken  as  a  perfectly  general  one 
and,  in  the  same  way,  the  instantaneous  motion  of  the  point  P  is  assumed  as 
indicated  by  the  arrow.  By  methods  indicated  in  earlier  chapters  locate  the 
centres  de  and  bd.  Continue  the  lines  ad  —  ab  and  be  —  ab  until  they  cut  the  line 
P  —  de  at  x  and  y,  respectively.  It  is  necessary  to  prove  that  P'  will  always 


PARALLEL  OR  STRAIGHT-LINE  MOTIONS.  43 

35.  Grasshopper  Motion.  —  A  device  which  may  be  used  to 
change  the  direction  of  rectilinear  motion  through  a  right  angle 
is  the  linkage  known  as  the  Grasshopper  Motion,  shown  in  Fig.  28. 
This  is  the  ordinary  slider-crank  chain  with  crank  a  and  con- 
necting-rod b  of  equal  length.  The  linkage  is  further  modified 

move  in  a  path  parallel  to  P's  motion  and  at  a  constant  proportion  to  it.  If 
this  is  true  for  instantaneous  motion  it  is  true  for  any  motion  P  may  be  given 
relatively  to  d.  Draw  the  line  P'  —  bd.  It  can  be  shown  that  this  line  will 
always  be  parallel  to  P  —  de,  for,  since  ac  —  x  is  parallel  to  ce—P 

P-be  =  x-bd 
be—ce     bd  —  ac' 

P-be      P-P/ 

Also,  by  similar  triangles  -  -  =  ™  -  . 

be—ce     P'  —ac 

x-bd      P-P' 

Hence,  TJ  -  =•»>/  -  » 

bd—ac     P'  —ac 

which,  considering  the  triangles  ac—  x  —  P  and  ac  —  bd—P',  shows  thatP'  —  bd 
is  parallel  to  P—x,  or  to  P-de. 

But  P  is  a  point  of  e  and  as  such  has  an  instantaneous  motion  relative  to  d  in 
a  direction  perpendicular  to  P  —  de.  In  the  same  way,  P'  is  a  point  of  b  and  as 
such,  relatively  to  d,  has  instantaneous  motion  perpendicular  to  P'  —  bd.  These 
two  instantaneous  motions  are  therefore  parallel. 

It  remains  to  be  shown  that  they  will  always  be  in  the  same  proportion  as  to 
extent.  The  extent  will  be  directly  proportional  to  the  instantaneous  linear 
velocities.  Both  P  and  be  as  points  of  e  rotate  for  the  instant  about  de 

VIP     P-de 

relatively  to  a,     .'.  TrT  —  =;  -  ~. 
VI  be     be-  de 

Both  be  and  P'  are  points  of  b,  and  as  such,  relatively  to  d,  rotate  about  bd, 
VI  be      be-bd     be-de  _ 

•••  vn*=w=bd=^d-e  (by  Slmilar  trlangles)- 

Multiplying, 

VIP      Vibe      P-de      be-de  VIP     P-de 

=  ^  similar  triangles), 

p  _ 


=  - 
be—ce 


a  constant  value. 


Or,  in  other  words,  the  linear  velocities  of  P  and  P'  bear  a  constant  ratio  to  each 
other  for  all  positions  of  the  mechanism,  and  hence,  these  points  will  trace 
proportionately  similar  paths  on  d.  —  Q.E.D. 


44 


MACHINE  DESIGN. 


by  extending  b  beyond  ab  to  a  point  P  such  that  the  length 
P-ab  =  crank  length.  It  is  obvious  that  P  is  constrained  to  move 
relative  to  d  in  a  straight-line  path  perpendicular  to  d  through 


36.  General  Method  for  Parallel-motion  Design.  —  A  general 
method  of  design  which  is  applicable  in  many  cases  is  as  follows. 
In  Fig.  29  d  is  the  fixed  link,  and  a  is  connected  with  it  by  a  sliding 
pair,  a,  b,  c,  and  e  are  connected  by  turning  pairs,  as  shown. 
The  constrainment  is  not  complete  because  B  is  free  to  move 
in  any  direction,  and  its  motion  would,  therefore,  depend  upon 
the  force  producing  it.  It  is  required  that  the  point  B  shall  move 


/f 

e 

6 

c       c 

t 

p 

/\Q 

T 

e 

b 

y' 

\. 

>  

a{ 

r- 

T 

FIG.  29. 

in  a  straight  line  parallel  to  a.  Suppose  that  B  is  caused  to  move 
along  the  required  line;  then  any  point  of  the  link  c,  as  ^4,  will 
describe  some  curve,  FAE.  If  a  pin  be  attached  to  c,  with  its 
axis  at  A,  and  a  curved  slot  fitting  the  pin,  with  its  sides  parallel 
to  FAE,  be  attached  to  d,  as  in  Fig.  30,  it  follows  that  B  can  only 
move  in  the  required  straight  line.  This  is  the  mechanism  of  the 
Tabor  Steam-engine  Indicator. 

*  This  is  true  because,  from  the  construction  of  the  mechanism,  the  line  P-bd 
must  always  lie  parallel  to  d.  The  point  P,  which  rotates  about  bd  as  center  rela- 
tively to  d,  always  has  an  instantaneous  motion  perpendicular  to  P-bd  and,  con- 
sequently, perpendicular  to  d. 


PARALLEL    OR  STRAIGHT-LINE  MOTIONS. 


45 


The  curve  described  by  A  might  approximate  a  circular  arc 
whose  center  could  be  located,  say,   at  O,  Fig.   30.     Then  the 


curved  slot  might  be  replaced  by  a  link  attached  to  d  and  c  by 
turning  pairs  at  O  and  A.  This  gives  B  approximately  the 
required  motion.  This  is  the  mechanism  of  the  Thompson 
Steam-engine  Indicator. 

If,  while  the  point  B  is  caused  to  move  in  the  required  straight 
line,  a  point  in  b,  as  P,  Fig.  29,  were  chosen,  it  would  be  found 


FIG.  31. 

to  describe  a  curve  which  would  approximate  a  circular  arc, 
whose  center,  O,  and  radius,  =r,  could  be  found.     Let  the  link 


46  MACHINE  DESIGN 

whose  length  =  r  be  attached  to  d  and  b  by  turning  pairs  whose 
axes  are  at  O  and  P,  and  the  motion  of  B  will  be  approximately 
the  required  motion.  This  is  the  mechanism  of  the  Crosby 
Steam-engine  Indicator.  One  very  important  fact,  however, 
is  to  be  noted  in  connection  with  all  steam-engine  indicator 
pencil  mechanisms.  While  it  is  important  that  the  pencil  point 


ac3 


ded 


FIG.  304. 

B  (Figs.  29  and  30)  travel  in  a  straight-line  path  parallel  to  the 
axis  of  the  piston  rod  a,  it  is  fully  as  important  that  the  motion 
of  the  point  B  always  be  exactly  the  same  multiple  of  a's  motion. 
To  determine  in  any  given  case  whether  this  is  true  or  not,  lay 
off  very  accurately  and  to  a  large  scale,  say  five  times  actual  size, 
a  skeleton  outline  of  the  mechanism  for  three  positions.  See 
Fig.  30^4 .  These  positions  are  taken  so  that  the  total  distance 


PARALLEL   OR  STRAIGHT-LINE  MOTIONS.  47 

Bl  —  B3  represents  the  allowable  range  of  the  instrument  as 
stated  by  the  maker,  usually  about  3".  B2  is  located  at  the  mid- 
position.  The  links  a,  b,  c,  e  and  /are  drawn  for  each  case  in 
their  proper  relative  positions,  d  being  considered  as  the  fixed 
link. 

The  subscripts  i,  2  and  3  refer  to  the  positions  of  the  links 
corresponding  to  the  three  pencil  positions  Bv  B2  and  By 

First  take  position  Br  The  centros  df  and  de  are  located 
at  once  because  they  are  permanent  centers  as  well.  Since  #'s 
motion  relative  to  d  is  rectilinear  translation,  the  centro  ad  will 
lie  at  infinity  in  a  direction  at  a  right  angle  to  the  direction  of 
motion,  or,  in  this  case,  at  horizontal  infinity.  The  centros  cfv 
abv  bcv  and  cel  are  located  at  once  at  the  axes  of  the  turning 
pairs  connecting  the  respective  links. 

Using  Kennedy's  theorem  locate  cdl  (on  lines  de—cel  and 
df—cfi),  and  acl  (on  lines  cd^  —  ad  and  abl  —  bcl).  At  the 
instant  in  question,  every  point  of  c  relatively  to  d  is  rotating 
about  the  centro  cdt  and  each  point  will  have  a  linear  velocity 
proportional  to  its  distance  from  cdr  But  B^  and  ac1  are  both 

.  .     >  .,     Vlac.       ac.  —  cd 

points  of  c.     Hence  we  may  write  *  =  —  '  -  -'  •       But    ac. 


is  also  a  point  of  a  and  at  any  instant  every  point  of  a  has  the 
same  velocity  relatively  to  d  because  the  motion  of  a  relative  to  d 

.,.  |     .  VI  a       ac.  —  cd. 

is  rectilinear  translation  .'.  -         =  —  l  -  *• 

VIB,      B,  -cdl 

Similarly  for  the  second  position, 

VI  a    _  ac2—  cd^ 
VI  B2  =  B2-  cd2  * 

And  for  the  third  position, 

VI  a    _  acH—  cd? 
VI  B,  ~B3-cd3  ' 


MACHINE  DESIGN. 


But,  for  proper  action, 

Via         Via         Via 
VI  B,  ~  VI  B2       VI B3 


=  a  constant  for  all  positions, 


,.,          ,  ac2  —  cd2  .  ac.,  — 

should  equal  — -2 — y-2,    and  also  equal  — — 
Bl  —  cd^  Bi  —  cdv  B3  — 

otherwise  the  diagrams  will  give  a  distortion  of  the  piston,  a's, 
motion.  Also  for  true  parallel  motion  Bv  should  lie  on  the  same 
horizontal  through  cdl  on  which  acl  lies;  B2  on  the  horizontal 
through  cd2-,  and  B3  on  the  horizontal  through  cdy 

An  examination  of  existing  indicator  mechanisms  in  this 
manner  gives  very  interesting  results,  and  separates  clearly 
those  instruments  which  distort  from  those  which  are  correct. 

37.  Problem.  —  In  Fig.  31,  B  is  the  fixed  axis  of  a  counter- 
shaft; C  is  the  axis  of  another  shaft  which  is  free  to  move  in 
any  direction.  It  is  required  to  constrain  D  to  move  in  the 
straight  line  EF.  If  D  be  moved  along  EF,  a  tracing-point 
fixed  at  A  in  the  link  CD  will  describe  an  approximate  circular 
arc,  HAK,  whose  center  may  be  found  at  O.  A  link  whose 
length  is  OA  may  be  connected  to  the  fixed  link,  and  to  the  link 
CD,  by  means  of  turning  pairs  at  O  and  A.  D  will  then  be 
constrained  to  move  approximately  along  EF.  A  curved  slot 
and  pin  could  be  used,  and  the  motion  would  be  exact. 

37A.  Peaucellier  Straight-line  Motion. — This  is  an  eight- 
link  chain,  all  link  connections  being  turning  pairs,  which  gives 
true  straight-line  motion  within  the  limits  of  its  action.  The 
mechanism  is  shown  in  two  positions  in  Fig.  31^!,  one  in  heavy 
lines,  the  other  in  light  lines.  The  fixed  link  is  d,  and  when  a 
is  oscillated  about  the  centro  ad,  the  point  P  travels  in  a  straight 
line.  The  linkage  is  symmetrical:  a  =  d,  b  =  c,  and  e,  f,  g,  and 
h  are  equal.  Because  of  this  symmetry,  whatever  position  of 
the  mechanism  be  considered,  A  and  P  will  lie  in  a  straight  line 
passing  through  D,  as  will  also  the  mid-point  O  of  the  diagonal 
A-P.  Since  e,  f,  g,  and  h  are  equal,  it  can  be  proven  readily 


PARALLEL    OR  STRAIGHT-LINE   MOTIONS. 


49 


that  the  diagonal  B-C  bisects  A-P  at  O  and  makes  an  angle 
of  90°  with  it. 
Therefore, 


and 


=  CO2+PO2, 


DC  -CP  = 


-PO2, 


=  (DO-PO)(DO+PO), 
=  DAxDP. 

DC  and  CP,  .being  the  length  of  the  respective  links  c  and  /? 
are  constant  for  the  mechanism  irrespective  of  position;    from 


which  it  follows  that  the  first  term  in  the  foregoing  equation 
is  a  constant  and,  therefore,  the  product  of  the  variable  distances 
DA  and  DP  is  a  constant. 

Consider  a  to  have  swung  to  the  position  ad- A'.  The 
mechanism  will  then  be  constrained  to  assume  the  position  shown 
by  the  light  lines.  From  what  has  just  been  proven, 

DP  =  DA'XDP', 
DA      DP' 


50  MACHINE  DESIGN. 

The  triangles  DPP'  and  DA' A,  having  an  angle  in  common 
with  its  adjacent  sides  proportional,  are  similar.  But  the  end 
of  link  a  in  swinging  from  position  A  to  A'  travels  in  the  cir- 
cumference of  a  circle  whose  diameter  =  DA  (d  and  a  being  equal), 
and  consequently  the  angle  DA' A  is  a  right  angle.  From  this 
it  follows  that  the  angle  DPP'  must  also  be  a  right  angle.  The 
proof,  being  independent  of  the  position  chosen,  holds  for  all 
positions  and,  therefore,  the  point  P  travels  in  a  straight  line 
perpendicular  to  DP. 

The  theoretical  limits  to  this  motion  are  the  positions  on 
either  side  of  the  line  DP  when  DP'=(c+f)* 

*  Descriptions  of  many  varieties  of  parallel  motions  maybe  found  in  Rankine's 
"Machinery  and  Millwork";  Weisbach's  "Mechanics  of  Engineering,"  Vol.  Ill, 
"Mechanics  of  the  Machinery  of  Transmission";  Kennedy's  "Mechanics  of 
Machinery";  and  elsewhere. 


CHAPTER  IV. 

CAMS. 

38.  Cams  Defined. — A  cam  is  a  machine  part  of  irregular 
outline   which,   having   a  motion   of  reciprocation,   rotation  or 
oscillation,    communicates    motion   by   means   of   point   or  line 
contact  to  another  part,  called  the  follower.     In  most  cases  the 
follower  is  to  be  given  a  motion  of  reciprocation  or  oscillation 
according  to  stated  conditions  governing  the  positions  it  is  to 
occupy  for  definite  positions  of  the  cam. 

The  method  of  design  is  the  same  in  all  cases,  and  consists 
in  laying  down  in  the  plane  of  the  cam  the  successive  positions 
which  the  follower  surface  is  to  occupy  relative  to  it,  and  then 
in  drawing  the  outline  of  the  cam  tangent  to  these  positions. 
It  follows  from  this  that  the  form  of  the  cam  outline  depends 
not  only  on  the  law  of  motion  which  the  follower  is  to  obey, 
but  also  depends  upon  the  form  of  the  contact  surface  of  the 
follower.  These  principles  can  be  understood  most  readily  by 
considering  a  few  simple  cases. 

39.  Case  i. — A  cam — which   rotates   counterclockwise,   with 
uniform   angular  velocity,   about   a  given   center — is  to  impart 
a  motion  of  straight-line  reciprocation  to  a  follower,  the  center 
line  of  whose  action  passes  through  the  cam  center.     The  fol- 
lower is  to  rise  with  uniform  linear  velocity  from  its  lowest  to 
its  highest  position  while  the  cam  rotates  through  135°;   it  is  to 
remain  at  rest  while  the  cam  rotates  through  the  next  45°;    and 
it  is  to  return  with  uniform  velocity  to  its  lowest  position  while 
the  cam  completes  its  rotation.     Three  cams  will  be  considered: 
(a)  the  follower  is  pointed,  i.e.,  wedge-shaped;    (b)  the  follower 

51 


MACHINE  DESIGN. 


has  a  cylindrical  roller;    and  (c)  the  follower  has  a  flat  contact 

surface  normal  to  its  line  of  action. 

Consider  first  the  case  with  the  pointed  follower  shown  in 

Fig.  32.     With  a  center  at  O,  the  center  of  rotation  of  the  cam, 

a  circle  is  drawn  through  the  point 
occupied  by  the  tip  of  the  follower  in 
its  lowest  position.  This  is  known  as 
the  base  circle.  This  circle  is  now 
divided,  in  a  clockwise  direction 
(against  the  direction  of  cam  rotation), 
into  three  major  portions,  of  135°,  45°, 
and  1 80°,  respectively,  corresponding 
to  the  major  divisions  of  the  follower's 
cycle  of  motion.  The  path  of  the 
follower  point  is  next  divided  into 
any  convenient  number  of  parts,  say 


FIG.  32. 


three,  equal  in  this  case  since  the  follower  is  to  rise  uniformly. 
The  more  divisions  there  are  taken,  the  more  accurately  will 
the  cam  outline  be  determined.  The  135°  portion  of  the  base 
circle  is  divided  into  the  same  number  of  equal  parts  by  radial 
lines.  Beginning  with  the  lowest,  number  the  follower  point 
positions  i,  2,  3,  and  4,  on  the  right-hand  side  of  the  line  of 
action.  Beginning  with  the  vertical  radial  line  of  the  cam,  num- 
ber these  elements  similarly  i,  2,  3,  and  4.  When  element  i  of 
the  cam  is  in  the  vertical  position  the  follower  tip  is  to  be  at  the 
distance  O-i  from  the  cam  center.  When  element  2  of  the  cam 
has  swung  up  to  the  vertical  position,  the  follower  point  is  to  be 
at  the  distance  O-2  from  the  cam  center.  With  this  distance 
as  radius,  and  O  as  center,  swing  an  arc  cutting  the  cam  element 
2  and  at  this  point  of  intersection  draw  the  follower  tip,  showing 
in  the  plane  of  the  cam  the  simultaneous  relation  of  cam  and 
follower  when  element  2  reaches  the  vertical  position.  Repeat 
this  construction  for  elements  3  and  4.  Since  the  follower  remains 


CAMS. 


stationary  for  the  next  45°  of  the  cam's  rotation,  the  relative 
positions  of  cam  element  and  follower  surface  are  the  same  for 
element  5  as  for  4.  Divide  the  remaining  180°  of  the  cam  into 
as  many  equal  parts  as  may  be  convenient,  say  4,  numbering 
the  successive  radial  lines  5,  6,  7,  8,  and  i.  Similarly  divide  the 
distance  the  follower  is  to  descend  during  this  angle  of  rotation 
of  the  cam  into  4  equal  parts,  numbering  the  respective  posi- 
tions of  the  follower  point  5,  6,  7,  8,  and  i,  on  the  left-hand  side 
of  the  line  of  action.  (If  the  follower  were  not  to  descend  with 
uniform  velocity,  these  divisions  would,  of  course,  no  longer  be 
made  equal,  but  would  be  laid  off  in  direct  proportion  to  the 
motion  to  be  imparted  to  it.)  Continue  the  layout  of  simul- 
taneous positions  for  this  portion  of  the  cam.  Draw  a  smooth 
curve  through  the  positions  occupied 
by  the  follower  points  on  the  radial 
lines — the  portion  from  elements  4 
to  5  being  a  circular  arc.  This 
gives  the  outline  of  the  cam  which 
will  give  the  desired  motion  to  the 
follower. 

Example  (b)  is  constructed  in 
exactly  the  same  manner.  See  Fig. 
33.  In  this  case  the  base  circle  is 
drawn  through  the  lowest  position 
of  the  follower  roller  center.  The 
positions  which  this  center  is  to 
occupy  for  the  various  angles  of  cam  rotation  are  located,  just 
as  were  the  pointed  follower  tips,  and  swung  to  the  respective 
radial  lines.  At  each  center  thus  located  a  circle,  the  size  of  the 
follower  roller,  is  drawn  and  the  cam  outline  completed  by 
drawing  a  smooth  curve  tangent  to  these  circles. 

Example  (c),  Fig.  34,  shows  the  identical  construction  applied 
to  the  flat-faced  follower — the  successive  simultaneous  positions 


FIG.  33. 


54 


MACHINE  DESIGN. 


of  cam  element  and  lower  surface  of  follower  being  laid  down 
and  the  cam  outline  drawn  by  constructing  a  smooth  tangent 
curve  to  the  follower  surfaces.  This  type  of  follower  frequently 

calls  for  impossible  cams,  unless  the 
given  law  of  motion  maybe  mod- 
ified slightly. 

It  will  be  noted  that,  although 
the  law  of  motion  for  the  follower 
is  the  same  in  all  three  cases,  the 
form  of  the  cam  outline  derived 
varies  with  the  form  of  contact  sur- 
face of  the  follower. 

In  the  cases  considered  above 
the  follower  is  given  a  uniform  mo- 
tion up  and  down.  This  is  done  for 

simplicity.  As  a  usual  rule,  particularly  for  high-speed  operation, 
the  follower  neither  should  be  started  nor  stopped  abruptly. 
It  should  not  be  given  a  uniform  motion  throughout  its  path, 
but  should  be  given  a  gradually  accelerated  motion  at  the  start 
in  each  direction  and  a  similarly  retarded  motion  at  the  finish. 

B 
D 


FIG.  34. 


(a) 


FIG.  35. 


In  Fig.  35  (a)  there  is  shown  a  method,  which  explains  itself, 
for  so  dividing  the  path,  AB,  of  the  follower  that  the  latter  will 
be  given  accelerations  and  retardations  corresponding  to  simple 
harmonic  motion.  In  Fig.  35  (6)  there  is  shown  a  construction 
for  conveniently  dividing  the  follower  path  AB  to  give  still  more 


CAMS. 


55 


gradual  starting  and  stopping  accelerations  and  retardations. 
The  law  is  that  of  gravity  acceleration.  Both  cases  show  sym 
metrical  motion  of  the  follower  about  the  mid-point,  C,  of  its 
path.  This  is,  obviously,  not  essential.  Fig.  35  (c)  shows  the 
necessary  modification  to  adapt  method  (a)  for  making  the  fol- 
lower have  its  maximum  velocity  at  any  point,  D,  of  its  path- 
The  method  is  equally  applicable  to  construction  (b). 

40.  Case  II. — The  follower  is  pointed  and  is  to  have  a  motion 
of  straight-line  reciprocation,  but  now  its  line  of  action  passes 
to  one  side  of,  and  not  through,  the  center  of  rotation  of  the 
cam.  The  line  may  be  in  any  position  relatively  to  the  center 
of  rotation  of  the  cam ;  hence  it  is  a  general  case.  The  point 
of  the  follower  which  bears  on  the  cam  is  constrained  to  move 
in  the  line  MN,  Fig.  36.  O  is  the  center  of  rotation  of  the  cam. 
About  O  as  a  center,  draw  a  circle 
tangent  to  MN  at  /.  Then  A,  B, 
C,  etc.,  are  points  in  the  cam, 
found  by  dividing  the  base  circle 
with  radial  lines  corresponding  to 
the  angles  through  which  the  cam 
is  to  rotate  while  the  follower  oc- 
cupies successive  positions,  accord- 
ing to  the  method  described  in  the 
preceding  section.  When  the  point 
A  is  at  /  the  point  of  the  follower 
which  bears  on  the  cam  must  be  at  A'\  when  B  is  at  /  the 
follower  point  must  be  at  B' ;  and  so  on  through  an  entire  revolu- 
tion. Through  A,  B,  C,  etc.,  draw  lines  tangent  to  the  circle. 
With  O  as  a  center,  and  OAf  as  a  radius,  draw  a  circular  arc 
A' A",  intersecting  the  tangent  through  A  at  A".  Then  A" 
will  be  a  point  in  the  cam  curve.  For,  if  A  returns  to  /,  AA" 
will  coincide  with  JA',  A"  will  coincide  with  A',  and  the  cam 
will  hold  the  follower  in  the  required  position.  The  same  process 


$6  MACHINE  DESIGN. 

for  the  other  positions  locates  other  points  of  the  cam  curve. 
A  smooth  curve  drawn  through  these  points  is  the  required 
cam  outline.  Often,  to  reduce  friction,  a  roller  attached  to 
the  follower  rests  on  the  cam,  motion  being  communicated 
through  it.  The  curve  found  as  above  will  be  the  path  of  the 
axis  of  the  roller.  The  cam  outline  will  then  be  a  curve  drawn 
inside  of,  and  parallel  to  the  path  of  the  axis  of  the  roller,  at 
a  distance  from  it  equal  to  the  roller's  radius.  The  path  of  the 


FIG.  37. 


FIG.  37^. 


point  of  contact  between  the  follower  and  the  cam  is  not  confined 
to  the  line  MN  if  a  roller  is  used. 

41.  Case  III — This  is  the  same  as  Case  I  (c),  except  that 
the  positions  of  the  follower  surface,  instead  of  being  parallel, 
converge  to  a  point,  O',  Fig.  37,  about  which  the  follower  vibrates. 
The  solution  is  the  same  as  in  Fig.  34,  except  that  the  angle 
between  the  lines  representing  the  corresponding  positions  of  the 
lower,  or  contact,  surface  of  the  follower,  and  the  radial  lines, 
instead  of  being  a  right  angle,  equals  the  angle  between  the 
corresponding  positions  of  the  follower  surface  and  the  vertical. 


CAMS. 


57 


In  these  cases  the  cam  drives  the  follower  in  only  one  direc- 
tion; the  force  of  gravity,  the  expansive  force  of  a  spring,  or 
some  other  force  must  hold  it  in  contact  with  the  cam.  To 
drive  the  follower  in  both  directions,  the  cam  surface  must  be 


\ 


0 


FIG.  38. 

double,  i.e.,  it  takes  the  form  of  a  groove  engaging  with  a  pin 
or  roller  attached  to  the  follower,  as  in  Fig.  37^. 

This  method  is  inclined  to  produce  excessive  wear.  A  better 
method  is  to  have  the  follower  provided  with  two  rollers  on 
opposite  sides  of  the  cam-shaft  engaging  complementary  cams. 
See  Fig.  38. 

Cam  A  is  designed  to  give  the  desired  motion  to  the  follower 
through  the  medium  of  roller  i.  Every  position  of  this  roller 


58  MACHINE  DESIGN. 

causes  roller  2  to  occupy  a  definite  position,  and  the  complementary 
cam  B  is  so  designed  as  to  correspond  to  these  positions  of  roller 
2.  Cam  B  is  rigidly  mounted  on  the  same  shaft  as  A,  so  that 


FIG.  38C. 


FIG.  38£. 


the  two  cams  have  no  motion  relative  to  each  other.  If  the  line 
of  action  of  the  follower  passes  through  the  center  of  the  cam. 
shaft  as  shown  in  Fig.  38,  it  becomes  a  very  simple  matter  to 
draw  the  outline  of  cam  J5;  all  that  is  necessary  is  to  keep  the 


CAMS.  59 

sum  of  the  radial  lengths  a  +  b  =  a  constant  =  the  distance  be- 
tween the  centers  of  rollers  i  and  2. 

This  method  is  also  applicable  to  cams  engaging  flat-faced 
followers.  In  this  case  the  complementary  cams  operate  on 
parallel  faces  of  a  follower  yoke  as  shown  in  Fig.  3&4. 

In  the  special  case  in  which  the  law  of  motion  of  the  fol- 
lower for  one-half  the  revolution  of  the  cam  may  be  the  reverse 
of  that  for  the  other  half-revolution,  it  is  not  necessary  to  use 
complementary  cams.  A  follower  with  two  rollers  can  then 
be  used  on  a  single  cam  as  shown  in  Fig.  38^.  One  half  the 
cam  is  first  designed  (see  full  lines)  to  give  the  follower  the  de- 
sired motion  in  one  direction.  The  other  half  of  the  cam  (see 
dotted  lines)  is  then  determined  by  the  condition  that  the  fol- 
lower rollers  must  always  be  the  same  distance  apart,  hence 
A'-B'=A-B,etc. 

The  same  construction  can  be  applied  to  a  forked  or  yoke 
follower,  as  shown  in  Fig.  38C,  the  distance  between  the  parallel 
tangents  being  uniformly  equal  to  A-B. 

42.  Case  IV. — To  lay  oat  a  cam  groove  on  the  surface  of  a 
cylinder  to  give  the  follower  a  motion  of  reciprocation  parallel 
to  the  cam  axis. — A,  Fig.  39,  is  a  cylinder  which  is  to  rotate 
continuously  about  its  axis.  B  can  only  move  parallel  to  the 
axis  of  A.  B  may  have  a  projecting  roller  to  engage  with  a 
groove  in  the  surface  of  A.  CD  is  the  axis  of  the  roller  in  its 
mid-position.  EF  is  the  development  of  the  surface  of  the  cylinder. 
During  the  first  quarter- re  volution  of  A,  CD  is  required  to  move 
one  inch  toward  the  right  with  a  constant  velocity.  Lay  off 
GH  =  i",  and  HJ  =  ±KF,  locating  /.  Draw  G/,  which  will  be  the 
middle  line  of  the  cam-groove.  During  the  next  half-revolution  of 
A  the  roller  is  required  to  move  two  inches  toward  the  left  with  a 
uniformly  accelerated  velocity.  Lay  off  JL=2ff,  and  LM  =  \KF. 
Divide  LM  into  any  number  of  equal  parts,  say  four.  Divide 
JL  into  four  parts,  so  that  each  is  greater  than  the  preceding  one 


6o 


MACHINE  DESIGN. 


by  an  equal  increment.  This  may  be  done  as  follows: 
3+4=10.*  Lay  of  from  /,  o.iJL,  locating  a;  then  O.2/Z,  from 
a,  locating  6;  and  so  on.  Through  a,  b,  and  c  draw  vertical 
lines;  through  m,  n,  and  o  draw  horizontal  lines.  The  in- 
intersections  locate  dy  e,  and  /.  Through  these  points  draw  the 
curve  from  /  to  M,  which  will  be  the  required  middle  line 
of  the  cam-groove.  During  the  remaining  quarter-revolution 
the  roller  is  required  to  return  to  its  starting  point  with  a 


FIG.  39. 

uniformly  accelerated  velocity.  The  curve  MN  is  drawn  in 
the  same  way  as  JM.  Using  the  line  GJMN  as  a  locus  of 
centers,  swing  in  circles  of  a  diameter  equal  to  that  of  the  fol- 
lower roller.  The  envelope  of  these  is  the  developed  cam  groove. 
This  groove  can  now  be  projected  back,  from  the  developed 
cylinder  to  the  cylinder  A,  by  the  ordinary  method  of  descriptive 

*  The  most  frequent  type  of  uniformly  accelerated  motion  is  that  of  falling 
bodies,  in  which  the  successive  distances  gone  through  in  equal  intervals  of  time 
are  in  the  ratios:  i,  3,  5,  7,  etc. 


CAMS. 


61 


geometry.     In  other  words,  wrap  EF  upon  the  cylinder  A,  and 
the  required  cam  groove  is  located. 

The  same  method  is  applicable  for  determining  a  cam  groove 
on  a  conical  surface  to  give  the  follower  a  motion  parallel  to 
the  cone  surface  element. 


FIG. 


The  fundamental  method  laid  down  in  this  chapter  is  capable 
of  indefinite  application.  Fig.  39,4  shows  a  method  by  which 
two  cams,  operating  simultaneously  on  two  attached  follower 
links,  can  cause  a  given  point  of  the  mechanism  to  trace  almost 
any  closed,  plane  path,  however  complex.  Nor  is  the  method 
confined  to  generating  plane  motion.  The  wide  usefulness  of 
cams  in  machines  requiring  intricate  motions  is,  therefore,  ap- 
parent. 


CHAPTER  V. 

ENERGY  IN  MACHINES. 

43.  The  subject  of  motion  and  velocity,  in  certain  simple 
machines,  has  been  treated  and  illustrated.  It  remains  now  to 
consider  the  passage  of  energy  through  similar  machines.  From 
this  the  solution  of  force  problems  will  follow. 

During  the  passage  of  energy  through  a  machine,  or  chain  of 
machines,  any  one,  or  all,  of  four  changes  may  occur. 

I.  The  energy  may  be  transferred  in  space.     Example. — En- 
ergy is  received  at  one  end  of  a  shaft  and  transferred  to  the  other 
end,  where  it  is  received  and  utilized  by  a  machine. 

II.  The  energy  may  be  converted  into  another  form.     Exam- 
ples.— (a)  Heat   energy   into  mechanical  energy   by   the   steam- 
engine  machine  chain,     (b)  Mechanical  energy  into  heat  by  fric- 
tion,     (c)  Mechanical    energy    into  electrical    energy,    as    in   a 
dynamo-electric    machine;    or    electrical  energy  into  mechanical 
energy  in  the  electric  motor,  etc. 

III.  Energy  is  the  product  of  a  force  factor  and  a  space  factor. 
Energy  per  unit  time,  or  rate  of  doing  work,  is  the  product  of  a 
force  factor  and  a  velocity  factor,  since  velocity  is  space  per  unit 
time.     Either  factor  may  be  changed  at  the  expense  of  the  other; 
i.e.,  velocity  may  be  changed,  if  accompanied  by  such  a  change 
of  force  that  the  energy  per  unit  time  remains  constant.     Cor- 
respondingly, force  may  be  changed  at  the  expense  of  velocity, 
energy  per  unit  time  being  constant.     Example. — A  belt  trans- 
mits 6000  foot-pounds  per  minute  to  a  machine.     The  belt  veloc- 
ity is  1 20  feet  per  minute,  and  the  force  exerted  is  50  Ibs.     Fric- 

62 


ENERGY  IN  MACHINES.  63 

tional  resistance  is  neglected.  A  cutting-tool  in  the  machine 
does  useful  work;  its  velocity  is  20  feet  per  minute,  and  the  re- 
sistance to  cutting  is  300  Ibs.  Then,  energy  received  per  minute 
=  120X50  =  6000  foot-pounds ;  and  energy  delivered  per  minute 
=  20X300  =  6000  foot-pounds.  The  energy  received  therefore 
equals  the  energy  delivered.  But  the  velocity  and  force  factors 
are  quite  different  in  the  two  cases. 

IV.  Energy  may  be  transferred  in  time.  In  many  machines 
the  energy  received  at  every  instant  equals  that  delivered.  There 
are  many  cases,  however,  where  there  is  a  periodical  demand  for 
work,  i.e.,  a  fluctuation  in  the  rate  of  doing  wcrk;  while  energy 
can  only  be  supplied  at  the  average  rate.  Or  there  may  be  a  uni- 
form rate  of  doing  work,  and  a  fluctuating  rate  of  supplying 
energy.  In  such  cases  means  are  provided  in  the  machine,  or 
chain  of  machines,  for  the  storing  of  energy  till  it  is  needed.  In 
other  words,  energy  is  transferred  in  time.  Examples. — (a)  In 
the  steam-engine  there  is  a  varying  rate  of  supplying  energy  dur- 
ing each  stroke,  while  there  is  (in  general)  a  uniform  rate  of  doing 
work.  There  is,  therefore,  a  periodical  excess  and  deficiency  of 
effort.  A  heavy  wheel  on  the  main  shaft  absorbs  the  excess  of 
energy  with  increased  velocity,  and  gives  it  out  again  with  re- 
duced velocity  when  the  effort  is  deficient,  (b)  A  pump  delivers 
water  into  a  pipe  system  under  pressure.  The  water  is  used  in  a 
hydraulic  press,  whose  action  is  periodic  and  beyond  the  capacity 
of  the  pump.  A  hydraulic  accumulator  is  attached  to  the  pipe 
system,  and  while  the  press  is  idle  the  pump  slowly  raises  the 
accumulator  weight,  thereby  storing  potential  energy,  which  is 
given  out  rapidly  by  the  descending  weight  for  a  short  time  while 
the  press  acts,  (c)  A  dynamo-electric  machine  is  run  by  a  steam- 
engine,  and  the  electrical  energy  is  delivered  and  stored  in  storage 
batteries,  upon  which  there  is  a  periodical  demand.  In  this  case, 
as  well  as  in  case  (b),  there  is  a  transformation  of  energy  as  well 
as  a  transfer  in  time. 


64  MACHINE  DESIGN. 

44.  Force  Problems. — Suppose  the  slider-crank  mechanism  in 
Fig.  40  to  represent  a  shaping-machine,  the  velocity  diagram  of 


FIG.  40. 

the  slider  being  drawn.  The  resistance  offered  to  cutting  metal 
during  the  forward  stroke  must  be  overcome.  This  resistance 
may  be  assumed  constant.  Throughout  the  cutting  stroke  there 
is  a  continually  varying  rate  oj  doing  work.  This  is  because  the 
rate  of  doing  work  =  resisting  force  (constant)  X  velocity  (vary- 
ing). This  product  is  continually  varying,  and  is  a  maximum 
when  the  slider's  velocity  is  a  maximum.  The  slider  must  be 
driven  by  means  of  energy  transmitted  through  the  crank  a.  The 
maximum  rate  at  which  energy  must  be  supplied  equals  the  maxi- 
mum rate  of  doing  work  at  the  slider.  Draw  the  mechanism  in 
the  position  of  maximum  velocity  of  slider;  *  i.e.,  locate  the  center 
of  the  slider-pin  at  the  base  of  the  maximum  ordinate  of  the  veloc- 
ity diagram,  and  draw  b  and  a  in  their  corresponding  positions. 
The  slider's  known  velocity  is  represented  by  yt  and  the  crank- 
pin's  required  velocity  is  represented  by  a  on  the  same  scale. 
Hence  the  value  of  a  becomes  known  by  simple  proportion.  The 
rate  of  doing  work  must  be  the  same  at  c  and  at  ab  (neglecting 
friction). f  Hence  Rv\=  Fv%t  in  which  R  and  1/1  represent  the 

*  It  is  customary  to  assume  the  slider's  position  for  this  condition  to  be  that 
corresponding  to  an  angle  of  90°  between  crank  and  connecting-rod.  This  is  not 
exactly  true,  but  is  a  sufficiently  close  approximation  for  the  ordinary  proportions 
of  crank  and  connecting-rod  lengths.  For  method  of  exact  determination  of 
slider's  position  see  Appendix. 

t  The  effect  of  acceleration  to  redistribute  energy  is  zero  in  this  position,  be- 
cause the  acceleration  of  the  slider  at  maximum  velocity  is  zero,  and  the  angular 
acceleration  of  b  can  only  produce  pressure  in  the  iournal  at  ud.  If  Ra  equals 


ENERGY  IN  MACHINES.  65 

force  and  velocity  factors  at  c\  and  F  and  v2  represent  the  tangen- 
tial force  and  velocity  factors  ab.  R  and  v1  are  known  from  the 
conditions  of  the  problem,  and  v2  is  found  as  above.  Hence  F  may 

Rvi 

be  found,  = =  force  which,  applied  tangentially  to  the  crank- 

^2 

pin  center,  will  overcome  the  maximum  resistance  of  the  machine. 
In  all  other  positions  of  the  cutting  stroke  the  rate  of  doing  work 
is  less,  and  F  would  be  less.  But  it  is  necessary  to  provide  driv- 
ing mechanism  capable  of  overcoming  the  maximum  resistance, 
when  no  fly-wheel  is  used.  If  now  F  be  multiplied  by  the  crank 
radius,  the  product  equals  the  maximum  torsional  moment  ( =  M) 
required  to  drive  the  machine.  If  the  energy  is  received  on  some 
different  radius,  as  in  case  of  gear  or  belt  transmission,  the  maxi- 
mum driving  force  =  M  -f-  the  new  radius.  During  the  return 
stroke  the  cutting-tool  is  idle,  and  it  is  only  necessary  to  overcome 
the  frictional  resistance  to  motion  of  the  bearing  surfaces.  Hence 
the  return  stroke  is  not  considered  in  designing  the  driving  mech- 
anism. When  the  method  of  driving  this  machine  is  decided  on, 
the  capacity  of  the  driving  mechanism  must  be  such  that  it  shall 
be  capable  of  supplying  to  the  crank-shaft  the  torsional  driving 
moment  M,  determined  as  above. 

This  method  applies  as  well  to  the  quick-return  mechanisms 
given.  In  each,  when  the  velocity  diagram  is  drawn,  the  vector 
of  the  maximum  linear  velocity  of  the  slider,  *=Li,  and  of  the 
constant  linear  velocity  of  the  crank-pin  center,  =  L2,  are  known, 
and  the  velocities  corresponding,  Vi  and  v2>  are  also  known,  from 
the  scale  of  velocities.  The  rate  of  doing  work  at  the  slider  and 

the  force  necessary  to  produce  acceleration  of  the  slider  mass  at  any  position  and 
Fa  the  force  necessary  at  the  crank  pin  to  produce  tangential  acceleration  of  the 
rotating  mass  (assuming  a  variable  velocity  of  the  crank-pin  as  well  as  slider), 
then  the  equation  in  its  most  general  form  will  be  (R+Ra}vi  =  (F+  Fa)v2. 
With  uniform  rate  of  rotation  of  the  crank  this  becomes  (R -\- RG}VI  =  Fv2) ',  and 
for  position  corresponding  to  maximum  velocity  of  slider,  as  above,  Rvi  =  Fv2. 


66  MACHINE  DESIGN. 

at  the  crank-pin  center  is  the  same,  friction  being  neglected. 
Hence  Rv\  =Fv2,  or,  since  the  vector  lengths  are  proportional 

to  the  velocities  they  represent,  RLi  =FL2',  and  F  =  —p — .     There- 
in 

fore  the  resistance  to  the  slider's  motion,  =R,  on  the  cutting 
stroke,  multiplied  by  the  ratio  of  linear  velocity  vectors,  -~,  of 

slider  and  crank-pin,  equals  F,  the  maximum  force  that  must  be 
applied  tangentially  at  the  crank-pin  center  to  insure  motion. 
F  multiplied  by  the  crank  radius  =  maximum  torsional  driving 
moment  required  by  the  crank-shaft.  If  R  is  varying  and  known, 
find  where  Rv,  the  rate  of  doing  work,  is  a  maximum,  and  solve 
for  that  position  in  the  same  way  as  above. 

Where  the  mass  to  be  accelerated  is  considerable  the  maxi- 
mum effort  will  be  called  for  at  the  beginning  of  each  stroke. 
If  there  is  a  quick  return  the  maximum  effort  will  come  at  the 
beginning  of  the  return  stroke.  A  planer  calls  for  about  twice 
as  much  power  at  the  beginning  of  its  return  stroke  as  it  does 
during  its  cutting  stroke, 

45.  Force  Problems,  Continued. — In  the  usual  type  of  steam- 
engine  the  slider-crank  mechanism  is  used,  but  energy  is  supplied 
to  the  slider  (which  represents  piston,  piston-rod,  and  cross- 
head),  and  the  resistance  opposes  the  rotation  of  the  crank  and 
attached  shaft.  In  any  position  of  the  mechanism  (Fig.  41), 
force  applied  to  the  crank-pin  through  the  connecting-rod  may 
be  resolved  into  two  components,  one  radial  and  one  tangential. 
The  tangential  component  tends  to  produce  rotation;  the  radial 
component  produces  pressure  between  the  surfaces  of  the  shaft - 
journal  and  its  bearing.  The  tangential  component  is  approxi- 
mately a  maximum  when  the  angle  between  crank  and  connecting- 
rod  equals  90°,*  and  it  becomes  zero  when  C  reaches  A  or  B. 
If  there  is  a  uniform  resistance  the  rate  of  doing  work  is  constant. 

*  See  foot-note  on  page  64. 


ENERGY  IN  MACHINES.  67 

Hence,  since  the  energy  is  supplied  at  a  varying  rate,  it  follows 
that  during  part  of  the  revolution  the  effort  is  greater  than  the 
resistance;  while  during  the  remaining  portion  of  the  revolution 
the  effort  is  less  than  the  resistance,  and  the  machine  will  stop 
unless  other  means  are  provided  to  maintain  motion.  A  "fly- 
wheel "  is  keyed  to  the  shaft,  and  this  wheel,  because  of  slight 


FIG.  41. 

variations  of  velocity,  alternately  stores  and  gives  out  the  excess 
and  deficiency  of  energy  of  the  effort,  thereby  adapting  it  to  the 
constant  work  to  be  done.* 

46.  Problem.  —  Given  length  of  stroke  of  the  slider  of  a  steam- 
engine  slider-crank  mechanism,  the  required  horse-power,  01 
rate  of  doing  work,  and  number  of  revolutions.  Required  the 
total  mean  pressure  that  must  be  applied  to  the  piston. 

Let       L  =  length  of  stroke  =  i  foot  ; 
HP  =  horse-power  =  20; 
N  =  strokes  per  minute  =  200; 
F  =  required  mean  force  on  piston. 
Then  N  XL  =  200  feet  per  minute=mean  velocity  of  slider  =  V. 

Now,  the  mean  rate  of  doing  work  in  the  cylinder  and  at  the 
main  shaft  during  each  stroke  is  the  same  (friction  neglected); 
hence  FV  =  HP  X  33000, 


*  See  Chaptt-r  XVI. 


68 


MACHINE  DESIGN. 


47.  Solution  of   Force  Problem   in    the    Slider-crank  Chain. 

— In  the  slider-crank  chain  the  velocity  of  the  slider  necessarily 
varies  from  zero  at  the  ends  of  its  stroke  to  a  maximum  value 
near  mid-stroke.  The  mass  of  the  slider  and  attached  parts 
is  therefore  positively  and  negatively  accelerated  each  stroke. 
When  a  mass  is  positively  accelerated  it  stores  energy;  and 
when  it  is  negatively  accelerated  it  gives  out  energy.  The  amount 
of  this  energy,  stored  or  given  out,  depends  upon  the  mass  and 
the  acceleration.  The  slider  stores  energy  during  the  first  part 
of  its  stroke  and  gives  it  out  during  the  second  part  of  its  stroke. 


M  w,o. 


\ 


\1 


FIG.  42. 

While,  therefore,  it  gives  out  all  the  energy  it  receives,  it  gives 
it  out  differently  distributed.  In  order  to  find  exactly  how  the 
energy  is  distributed,  it  is  necessary  to  find  the  acceleration 
throughout  the  slider's  stroke.  This  may  be  done  as  follows: 
Fig.  42,  A,  shows  the  velocity  diagram  of  the  slider  of  a  slider- 
crank  mechanism  for  the  forward  stroke,  the  ordinates  repre- 
senting velocities,  the  corresponding  abscissae  representing  the 

Av 
slider  positions.     The  acceleration  required  at  any  point    =~7~.i 

in  which  Av  is  the  increase  in  velocity  during  any  interval  of 


ENERGY  IN  MACHINES.  69 

time  J/,  assuming  that  the  increase  in  velocity  becomes  constant 
at  that  point.  Lay  off  the  horizontal  line  OP=MN.  Divide 
OP  into  as  many  equal  parts  as  there  are  unequal  parts  in  MN. 
These  divisions  may  each  represent  J/.  At  m  erect  the  ordinate 
mn=min\,  and  at  o  erect  the  ordinate  op=o\p\.  Continue 
this  construction  throughout  OP,  and  draw  a  curve  through  the 
upper  extremities  of  the  ordinates.  Fig.  42,  B,  is  a  velocity 
diagram  on  a  "time  base."  At  O  draw  the  tangent  OT  to  the 
curve.  If  the  increase  in  velocity  were  uniform  during  the  time 
interval  represented  by  Om,  the  increment  of  velocity  would  be 
represented  by  mT.  Therefore  mT  is  proportional  to  the  accel- 
eration at  the  point  O,  and  may  be  laid  off  as  an  ordinate  of  an 
acceleration  diagram  (Fig.  426").  Thus  Qa  =  mT.  The  divi- 
sions of  QR  are  the  same  as  those  of  MN\  i.e.,  they  represent 
positions  of  the  slider.  This  construction  may  be  repeated  for 
the  other  divisions  of  the  curve  B.  Thus  at  n  the  tangent  nT\ 
and  horizontal  nq  are  drawn,  and  qTi  is  proportional  to  the 
acceleration  at  n,  and  is  laid  off  as  an  ordinate  be  of  the  ac- 
celeration diagram.  To  find  the  value  in  acceleration  units 
of  Qa,  mT  is  read  off  in  velocity  units  =Av  by  the  scale 
of  ordinates  of  the  velocity  diagram.  This  value  is  divided 
by  M,  the  time  increment  corresponding  to  Om.  The  result 

Av 

of   this   division   —  =  acceleration   at   M   in   acceleration   units. 
At 

J/=the  time  of  one  stroke,  or  of  one  half  revolution  of  the  crank, 
divided  by  the  number  of  divisions  in  OP.  If  the  linear  velocity 
of  the  center  of  the  crank-pin  in  feet  per  second,  =v,  be  repre- 

MN 
sented  by  the  length  of  the  crank  radius  =  —  —  =  a,  then  the  scale 

of  velocities,  or  velocity  in  feet  per  second  for  i  inch  of  ordinate, 

=—  = .     D  is  the  actual  diameter  of  the  crank  circle,  N 

a      a6o 

is  the  number  of  revolutions  per  minute,  and  a  is  the  crank  radius 
measured  on  the  figure. 


70  MACHINE  DESIGN. 

The  determination  of  the  acceleration  curve,  by  means  of 
tangents  drawn  to  the  "time- base"  velocity  curve,  has  a  serious 
drawback.  The  tangent  lines  are  laid  down  by  inspection,  and  slight 
inaccuracy  in  their  location  and  construction  may  lead  to  consid- 
erable errors  in  the  ordinates  obtained  for  the  acceleration  curve. 

The  following  method  is  therefore  suggested  as  an  alternative. 

If  one  point  is  rotating  about  another  point  with  a  given 
instantaneous  velocity =v  and  a  radius  =  r,  the  instantaneous 

v2 
radial  acceleration  of  either  point  toward  the  other  =  — . 

Consider  the  slider-crank  chain  in  the  position  at  the  begin- 
ning of  the  forward  stroke  as  shown  in  Fig.  43^.  The  problem 
is  to  determine  the  acceleration  of  the  point  be  toward  ad.  Ac- 
celerations toward  the  right  will  be  considered  as  positive,  toward 
the  left  as  negative.  In  the  position  chosen  the  point  ab  is  mov- 
ing, relatively  to  both  links  d  and  c,  in  the  direction  of  the  arrow 
with  a  velocity =v,  the  uniform  velocity  of  ab  relatively  to  d. 

The  acceleration  of  be  toward  ad  is  always  made  up  of  two 
components,  namely,  the  acceleration  of  be  toward  ab  and  the 
acceleration  of  ab  toward  ad.  In  the  position  under  considera- 

v2 
tion  the  acceleration  of  be  toward  ab=~r  in  a  positive  direction. 

v2 

Similarly  the  acceleration  of  ab  toward  ad  =  —  in  a  positive  di- 
rection.    The  total  acceleration  of  be  toward  ad  therefore  equals 

v2     v2 
the  sum  of  these  two  components,  or  =  T+~- 

On  the  other  hand,  at  the  end  of  the  forward  stroke,  shown 

v* 

in  Fig.  43$,  the  acceleration  of  be  toward  ab  =  ~r  in  a  positive 

i2 

direction  as  before,  while  the  acceleration  of  ab  toward  ad=— 

a 

in  a  negative  direction.     The  algebraic  sum  of  these  two  com- 


ENERGY  IN  MACHINES. 


V2      V2 


71 


ponents   therefore  =  -r  —  — .     This   quantity   will   always   have   a 

negative  value,  since  in  the  slider-crank  mechanism  a  must  al- 
ways be  smaller  than  b. 

To  construct  the  acceleration  curve,  lay  off  a  length  MN 


FIG.  44. 


(Fig.  43Q  proportionate  to  the  length  of  the  stroke  of  the  slider. 

v2    v2 
At  M  erect  an  ordinate,  MP,  whose  value  equals  T~  +  ~-     It 

is  best  to  use  for  these  ordinates  a  scale  on  which  a  (the  length 


MACHINE  DESIGN. 


of  the  crank)  represents  the  value  — .     At  N  erect  the  negative 

v2    v2 

ordinate  NQ=-r— — .* 
^      b      a 

There  is  a  position  of  the  slider,  O,  where  the  acceleration 
equals  zero.     This  must  correspond  to  the  position  of  the  slider 

*  The  following  construction  for  graphically  obtaining  ordinates  representing 

V2  V2  V2          V2 

—  +  —  and j  on  the  scale  upon  which  a  represents  v 

b         a          b        a 

(and,   hence,  a  =  —  )  is  due  to  Professor  Le  Conte. 
a) 


FIG.  44^. 

Reference  is  to  Fig.  44^.  M  and  N  represent  the  position  of  the  slider  at  the 
beginning  and  end  of  the  stroke,  respectively. 

At  abl  erect  the  vector  v  ( =  a)  and  from  M  draw  a  line  through  its  upper  extremity. 
Prolong  this  line  until  it  cuts  the  perpendicular  through  ad,  thus  determining  the 
length  yr 

At  ab2  lay  off  downward  the  vector  v  (  =  #).  From  N  draw  a  line  to  the  lower 
extremity  of  this  vector,  cutting  off  the  length  y2  on  the  perpendicular  through  ad 

Then  will  y1  represent—  H — ;    and  yz  represent  r • 

For,  taking  slider  position  M,  by  similar  triangles, 
a2  +  ab      i 


For  position  AT,  by  similar  triangles, 

a2  —    ab      V2 


LngleS'a  +  6           b 

x.       But  a  =  -,    .-.  ^  =  r  H 
a                     b 

,2? 
a 

V2    _  v  (  =  a) 

a-b           b 

ENERGY  IN  MACHINES.  73 

when  it  has  its  maximum  velocity,  which  may  be  taken  from 
the  original  velocity  diagram  of  the  slider,  or,  with  greater 
accuracy,  from  Curve  B  in  the  Appendix.  Through  POQ  draw 
a  smooth  curve.  For  most  purposes  this  curve  will  be  accurate 
enough. 

Where  more  points  of  the  curve  are  desired  for  the  sake  of 
greater  accuracy  the  method  illustrated  in  Fig.  44  may  be 
used.  Assume  the  slider  in  the  position  at  which  its  acceleration 
is  desired  and  draw  the  crank  a  and  connecting-rod  b  in  their 
corresponding  positions.  Locate  the  centros  ad,  ab,  ac,  and  bd. 
From  ac  draw  a  parallel  to  be— ad  until  it  cuts  the  crank,  pro- 
longed if  necessary,  at  A.  From  A  draw  a  parallel  to  ad-ac 
until  it  cuts  the  connecting-rod  at  B.  From  B  draw  a  perpen- 
dicular to  the  connecting-rod  until  it  cuts  bc-ad,  prolonged  if 
necessary,  at  C.  Then  ad-C  is  the  desired  ordinate  of  the 

v2 
acceleration  diagram  on  the  scale  by  which  the  length  a  =  —. 

The  proof  is  as  follows,  reference  being  made  to  Fig.  44. 

At  this  instant  every  point  of  b  relatively  to  c  is  swinging 
about  the  centro  be  with  a  velocity  proportional  to  its  distance 
from  be. 

vel.  of  bd  rel.  to  c     bd-bc     ac-ad 
vel.  of  ab  rel.  to  c     ab-bc     ac-ab' 

But  ac-ad  represents  the  velocity  of  c  relatively  to  d  (or  d 
relatively  to  c)  on  the  same  scale  that  ad-ab  represents  the 
velocity  of  the  point  ab  relatively  to  d.  Therefore  ac-ab  repre- 
sents the  velocity  of  ab  rotating  about  be  relatively  to  c  on  the  same 
scale  that  ad-ab  represents  the  velocity  of  ab  relatively  to  d. 

Hence  the  radial  acceleration  of  ab  toward  be  *  (or  conversely 

*  The  acceleration  of  a  point  A  with  respect  to  another  point  B  is  the  accel- 
eration of  A  with  respect  to  a  non-rotating  body  of  which  B  is  a  point. 


74  MACHINE  DESIGN. 


of  be  toward  ab)  =  - — 7 — ,  which  is   represented  by  the  length 

ab-B,  as  can  be  shown  as  follows: 
By  similar  triangles 

ab-B     ab-A      ab-ac 
ab-ac     ab-ad    ab-bc' 


7    Tf     ab-ac      ab-ac 
..  ab-B  — 


ab-bc          b 

_  _o 

The    radial    acceleration    of   ab    toward     ad=^—     —  ,  whose 

a 

value  we  represent  by  the  length  a.  The  component  of  this 
acceleration  in  the  direction  bc-ab=ab-D. 

The  acceleration  of  be,  relatively  to  d,  along  the  path  bc-ab 
is  made  up  of  two  components:  ist,  the  acceleration  of  be  toward 
ab(=B-ab)  plus,  2d,  the  acceleration  of  ab  relatively  to  d  along 
the  same  path  (=ab-D). 

In  the  position  shown  this  algebraic  sum  is  the  negative 
quantity  represented  by  B-D.  But  the  actual  direction  of  bc's 
acceleration  relatively  to  d  is  along  the  line  bc-ad.  Its  accelera- 
tion. in  this  direction  must  therefore  be  the  quantity  whose  com- 
ponent along  ab-bc  is  B-D,  namely,  C-ad.  Q.E.D. 

If  the  weight  W  of  parts  accelerated  is  known,  the  force  F 
necessary  to  produce  the  acceleration  at  any  slider  position  may 
be  found  from  the  fundamental  formula  of  mechanics, 


p  being  the  acceleration  corresponding  to  the  position  considered. 
If  the  ordinates  of  the  acceleration  diagram  are  taken  as  repre- 
senting the  jorces  which  produce  the  acceleration,  the  diagram 
will  have  force  ordinates  and  space  abscissae,  and  areas  will 
represent  work.  Thus,  Qas,  Fig.  426*,  represents  the  work  stored 


ENERGY  IN  MACHINES.  75 

during  acceleration,  and  Rs d  represents  the  work  given  out  during 
retardation.  Let  MAT,  Fig.  45,  represent  the  length  of  the 
slider's  stroke  and  NC  the  resistance  of  cutting  (uniform)  on  the 
same  force  scale  as  that  by  which  Qa,  Fig.  42  C,  represents  the 

Wp 
force  at  the  beginning  of  the  stroke ;  then  energy  to  do  cutting 

o 

per  stroke  is  represented  by  the  area  MBCN.  But  during  the 
early  part  of  the  stroke  the  reciprocating  parts  must  be  acceler> 
ated,  and  the  force  necessary  at  the  beginning,  found  as  above, 
=  BD=Qa.  The  driving-gear  must,  therefore,  be  able  to  over- 
come resistance  equal  to  MB  +  BD.  The  acceleration,  and  hence 
the  accelerating  force,  decreases  as  the  slider  advances,  becoming 
zero  at  E.  From  E  on  the  acceleration  becomes  negative,  and 
hence  the  slider  gives  out  energy  and  helps  to  overcome  the  resist- 
ance, and  the  driving-gear  has  only  to  furnish  energy  represented 
by  the  area  AEFN,  though  the  work  really 
c  done  against  resistance  equals  that  repre- 
sented  by  the  area  CEFN.  The  energy 
represented  by  the  difference  of  these  areas, 
FIG.  45-  =ACE,  is  that  which  is  stored  in  the 

slider's  mass  during  acceleration.  Since  by  the  law  of  con- 
servation of  energy,  energy  given  out  per  cycle  =  that  received, 
it  follows  that  area  ,4CE=area  DEB,  and  area  BCMN  = 
ADMN.  This  redistribution  of  energy  would  seem  to  modify 
the  problem  on  page  62,  since  that  problem  is  based  on  the 
assumption  of  uniform  resistance  during  cutting  stroke.  The 
position  of  maximum  velocity  of  slider,  however,  corresponds  to 
acceleration  =o.  The  maximum  rate  of  doing  work,  and  the 
corresponding  torsional  driving  moment  at  the  crank-shaft  would 
probably  correspond  to  the  same  position,  and  would  not  be 
materially  changed.  In  such  machines  as  shape rs,  the  accelera- 
tion and  weight  of  slider  are  so  small  that  the  redistribution  of 
energy  is  unimportant. 


F 


76 


MACHINE  DESIGN. 


48.  Solution  of  the  Force  Problem  in  the  Steam-engine  Slider- 
crank  Mechanism.  (Slider  represents  piston  with  its  rod,  and  the 
cross-head.) — The  steam  acts  upon  the  piston  with  a  pressure 
which  varies  during  the  stroke.  The  pressure  is  redistributed 
before  reaching  the  cross-head  pin,  because  the  reciprocating  parts 
are  accelerated  in  the  first  part  of  the  stroke,  with  accompanying 
storing  of  energy  and  reduction  of  pressure  on  the  cross-head 
pin;  and  retarded  in  the  second  part  of  the  stroke,  with  accom- 
panying giving  out  of  energy  and  increase  of  pressure  on  the 
cross-head  pin.  Let  the  ordinates  of  the  full  line  diagram  above 
OX,  Fig.  46.4,  represent  the  total  effective  pressure  on  the  piston 
throughout  a  stroke.  Fig.  46^  is  the  velocity  diagram  of  slider. 


FIG.  46. 

Find  the  acceleration  throughout  stroke,  and  from  this  and  the 
known  value  of  weight  of  slider  find  the  force  due  to  acceleration. 
Draw  diagram  Fig.  466*,  whose  ordinates  represent  the  force 
due  to  acceleration,  upon  the  same  force  scale  used  in  A.  Lay 
off  this  diagram  on  OX  as  a  base  line,  thereby  locating  the  dotted 
line.  The  vertical  ordinates  between  this  dotted  line  and  the 
upper  line  of  A  represent  the  pressure  applied  to  the  cross-head 
pin.  These  ordinates  may  be  laid  off  from  a  horizontal  base  line, 
giving  D.  The  product  of  the  values  of  the  corresponding  ordinates 
of  B  and  D  =the  rate  oj  doing  work  throughout  the  stroke.  Thus 
the  value  of  GH  in  pounds  X  value  of  EF  in  feet  per  second  =the 


ENERGY  IN  MACHINES.  77 

rate  of  doing  work  in  foot-pounds  per  second  upon  the  cross- 
head  pin,  when  the  center  of  the  cross-head  pin  is  at  E.  The 
rate  of  doing  work  at  the  crank-pin  is  the  same  as  at  the  cross- 
head  pin.  Hence  dividing  this  rate  of  doing  work,  =EFxGH, 
by  the  constant  tangential  velocity  of  the  crank-pin  center,  gives 
the  force  acting  tangentially  on  the  crank-pin  to  produce  rotation. 
The  tangential  forces  acting  throughout  a  half  revolution  of 
the  crank  may  be  thus  found,  and  plotted  upon  a  horizontal 
base  line=length  of  half  the  crank  circle  (Fig.  47$).  The  work 
done  upon  the  piston,  cross-head  pin,  and  crank  during  a  piston 
stroke  is  the  same.  Hence  the  areas  of  A  and  D,  Fig.  46,  are 
equal  to  each  other,  and  to  the  area  of  the  diagram,  Fig.  47$. 
The  forces  acting  along  the  connecting-rod  for  all  positions 
during  the  piston  stroke  may  be  found  by  drawing  force  triangles 
with  one  side  horizontal,  one  vertical,  and  one  parallel  to  position 
of  connecting-rod  axis,  the  horizontal  side  being  equal  to  the 
corresponding  ordinate  of  Fig.  46!).  The  vertical  sides  of  these 
triangles  will  represent  the  guide  reaction,  while  the  side  parallel 
to  the  connecting-rod  axis  represents  the  force  transmitted  by 
the  connecting-rod. 

The  tangential  forces  acting  on  the  crank-pin  may  be  found 
graphically  by  the  method  shown  in  Fig.  47^.  Let  GH  repre- 
sent the  net  effective  force  acting  in  a  horizontal  direction  at  the 
center  of  the  cross-head  pin. 

It  has  been  shown  that  EF  represents  the  velocity  of  the 
slider  on  the  same  scale  that  EA  represents  that  of  the  center  of 
the  crank-pin;  also  that  the  rate  of  doing  work,  after  having 
made  the  necessary  corrections  for  acceleration,  is  the  same  at 
the  center  of  the  crank-pin  as  at  the  slider,  i.e.,  GHXEF  =  tan- 
gential force  at  center  of  crank-pin  XEA.  Hence  the  tangential 

EF 
force  at  center  of  crank-pin  =  GH  X  TT-J  • 


MACHINE  DESIGN. 


Lay  off  AB=GH,  and  draw  BC  parallel  to  EF.     Then,  by 
similar  triangles, 

BC 

_ 


EF 

_, 


the  tangential  force  acting  at  the  crank-pin  center  for  the  assumed 
position  of  the  mechanism,  on  the  same  scale  as  GH"=net  effective 
horizontal  force  on  slider. 

Lay  off  AD=BC. 

Following  through  this  construction  for  a  number  of  positions 
of  the  mechanisms,  a  polar  diagram  is  determined  which  shows 


FIG.  475. 

very  clearly  the  relation  existing  between  the  varying  tangential 
forces  and  the  corresponding  crank  positions.  Before  this  dia- 
gram may  be  used  in  the  solution  of  the  fly-wheel  problem  (see 
Chapter  XVI)  it  should  be  transferred  to  a  straight-line  base 
whose  length  for  one  stroke  equals  the  semi-circumference  of  the 
crank-pin  circle.  That  is,  the  abscissae  will  be  the  distance  moved 
through  by  the  center  of  the  crank-pin  and  the  ordinates  will  be 
the  corresponding  radial  intercepts  AD.  The  diagram  so  ob 
tained  will  be  identical  with  that  shown  in  Fig.  47^. 


ENERGY  IN  MACHINES. 


79 


48A.  General  Method  for  Determining  Velocity  and  Accele- 
ration Diagrams. — The  construction  used  first  in  Sec.  47  for 
determining  the  acceleration  diagram  of  the  slider-crank  chain 
is  capable  of  wide  adaptation.  It  can  be  used  to  determine  not 
only  the  acceleration  diagram,  but  also  the  velocity  diagram  (upon 


A  «L 


Jft 


C  c 


D  c 


0  Q 

1  2 


FIG.  4yC. 

which  the  acceleration  diagram  is  based)  for  any  point  in  any 
body,  provided  only  that  the  path  of  the  point  be  known  together 
with  the  positions  it  occupies  at  the  end  of  equal  intervals  of 
time  throughout  its  cycle.  See  Fig.  476*.  At  A  is  shown  the 
path,  a-b,  the  point  traverses  in  the  given  time.  Starting  at 
i,  the  positions  2,  3,  4,  5,  6,  and  7  are  those  reached  after  sue- 


8o  MACHINE    DESIGN. 

cessive  equal  intervals  of  time.  At  B  a  convenient  distance, 
c-d,  is  taken  as  base-line  to  represent  the  time  required  by  the 
point  to  travel  the  distance  a-b.  c-d  is  divided  into  as  many 
equal  parts  as  a-b  has  unequal  parts,  and  these  distances  1-2, 
2-3,  etc.,  represent  equal  increments  of  time,  At.  At  2  on  c-d 
erect  the  ordinate  MN  =  distance  1-2  of  a-b.  Similarly  at  3  on 
c-d,  erect  an  ordinate  =  distance  1-3,  of  a-b.  Continue  for  all 
points  on  c-d.  The  ordinate  at  7  will,  of  course,  equal  a-b 
itself.  Draw  a  smooth  curve  through  the  ends  of  these  ordinates. 
This  will  give  a  displacement  curve  on  a  time  base.  The  ordinates 
represent  distances,  the  abscissae,  time.  At  N  draw  the  tan- 
gent NP  and  the  horizontal  NO.  Then  will  OP  represent  the 

velocity  of  the  point  when  it  is  at  position  2  ;  for  velocity  =  —  =  —  , 
and,  since  OP  equals  the  displacement  increment  for  the  time 
NOy  if  the  velocity  at  N  held  for  the  entire  interval  NO,  —  = 


OP  expressed  in  distance  units  divided  by  NO  in  time  units, 
gives  the  actual  velocity  of  the  point  at  position  2  in  velocity 
units.  It  is  clear  since  the  same  vector,  =7VO,  is  taken  each 
time  to  represent  At,  that  the  intercepts  OP,  TU,  etc.,  may  be 
taken  themselves  as  the  velocity  vectors  for  the  respective 
positions. 

At  C,  then,  which  is  the  velocity  diagram  of  the  point  on  a 
time  base,  OP  —  which  is  the  velocity  vector  for  position  2  —  is 
laid  off  as  an  ordinate  at  2.  Similarly  for  the  other  positions, 
as  TU  at  4.  The  smooth  curve  cPUd  is  then  drawn. 

From  the  velocity  diagram  on  the  time  base,  since  accelera- 

Av 
tion  =  —  ,  the  vectors  are  derived  for  the  acceleration  diagram 

as  described  in  sec.  47.  Such  an  acceleration  diagram  is  shown 
at  D,  laid  off  on  a  time  base.  It  could  be  transferred  readily 
to  the  position  base  shown  at  A. 


CHAPTER  VI. 

PROPORTIONS   OF  MACHINE   PARTS   AS   DICTATED   BY  STRESS. 

49.  The   size  and  form  of  machine  parts  *  are  governed  by 
six  main  considerations : 

(1)  The  size  and  nature  of  the  work  to  be  accommodated  (as 
the  swing  of  engine-lathes,  etc.). 

(2)  The  stresses  which  they  have  to  endure. 

(3)  The  maintaining  of  truth  and  accuracy  against  wear,  in- 
cluding all  questions  of  lubrication. 

(4)  The  cost  of  production. 

(5)  Appearance. 

(6)  Properties  of  materials  to  be  used. 

The  first  is  a  given  condition  in  any  problem ;  the  second  will 
be  discussed  here;  the  third  will  be  treated  in  the  chapters  on 
Journals  and  Sliding  Surfaces;  the  fourth  is  touched  upon  here; 
the  principles  governing  the  fifth  are  treated  in  Chapter  XIX 
and  here. 

It  is  assumed  in  this  and  following  chapters  that  the  reader  is 
familiar  with  the  properties  of  the  materials  employed  in  machine 
construction,!  and  with  the  general  principles  of  the  science  of 
mechanics.  A  few  tables  are  appended  to  this  chapter  for 
convenience. 

50.  The  stresses  acting  on  machine  parts  may  be  constant, 
variable,  or  suddenly  applied. 


*  On  this  general  subject  see  an  excellent  article  by  Prof.  Sweet  in  the  Jour- 
nal of  the  Franklin  Institute,  3d  Series,  Vol.  125,  pp.  278-300.  The  reader  is 
also  referred  to  "A  Manual  of  Machine  Construction,"  by  Mr.  John  Richards, 
and  to  the  Introduction  of  this  volume. 

f  See  Smith's  "Materials  of  Machines." 

81 


82  MACHINE  DESIGN. 

A  CONSTANT  stress  is  frequently  spoken  of   as  a  STEADY,  01 

DEAD,  LOAD. 

A  VARIABLE  stress  is  known  as  a  LIVE  LOAD. 

A  SUDDENLY  APPLIED  stress  is  known  as  a  SHOCK. 

51.  Constant  Stress. — If  a  machine  part  is  subjected  to  a  con- 
stant stress,  i.e.,  an  unvarying  load  constantly  applied,  its  design 
becomes  a  simple  matter,  as  the  amount  of  such  a  stress  can 
generally  be  very  closely  estimated.  Knowing  this  and  the 
properties  of  the  materials  to  be  used,  it  is  only  necessary  to  cal- 
culate the  area  which  will  sustain  the  load  without  excessive 
deformation. 

ThuSj  in  simple  tension  or  compression,  if  we  let  £7  =  the  ulti- 
mate strength  of  the  material  in  pounds  per  square  inch,  F  the 
total  constant  stress  in  pounds,  A  the  unknown  area  in  square 
inches  necessary  to  sustain  F,  we  write 


A  = 


U+K' 


where  K  is  a  so-called  FACTOR  OF  SAFETY,  introduced  to  reduce 
the  permitted  unit  stress  to  such  a  point  as  will  limit  the  deforma- 
tion (strain)  to  an  allowable  amount,  and  also  to  provide  for  pos- 
sible defects  in  the  material  itself.  In  exceptional  cases  where 
the  stresses  permit  of  accurate  calculation,  and  the  material  is  of 
proven  high  grade  and  positively  known  strength,  K  has  been 
given  as  low  a  value  as  i£;  but  values  of  2  and  3  are  ordinarily 
used  for  wrought  iron  and  steel  free  from  welds;  while  4  to  5  are 
as  small  as  should  be  used  for  cast  iron,  on  account  of  the  uncer- 
tainty of  its  composition,  the  danger  of  sponginess  of  structure, 
and  indeterminate  shrinkage  stresses. 

U 
The  SAFE  UNIT  STRESS  =f=—  in  pounds  per  square  inch. 

52.  Variable  Stress.  — We  pass  next  to  the  consideration  of 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.  83 

variable  stresses  or  live  loads.  Here  the  problem  is  much  more 
complex  than  with  dead  loads. 

Experiments  by  Wohler,*  and  Bauschinger,f  with  the  work 
of  Weyrauch  %  and  others  have  given  us  the  laws  of  bodies  sub- 
jected to  repeated  stresses.  In  substance  Wohler's  law  is  as 
follows:  MATERIAL  MAY  BE  BROKEN  BY  REPEATED  APPLICATIONS 
OF  A  FORCE  WHICH  WOULD  BE  INSUFFICIENT  TO  PRODUCE  RUPTURE 
BY  A  SINGLE  APPLICATION.  THE  BREAKING  IS  A  FUNCTION  OF 
RANGE  OF  STRESS;  AND  AS  THE  VALUE  OF  THE  RECURRING  STRESS 
INCREASES,  THE  -RANGE  NECESSARY  TO  PRODUCE  RUPTURE  DE- 
CREASES. IF  THE  STRESS  BE  REVERSED,  THE  RANGE  EQUALS 
THE  SUM  OF  THE  POSITIVE  AND  NEGATIVE  STRESS. 

Bauschinger's  conclusions  were  as  follows: 

(1)  WITH   REPEATED   TENSILE   STRESSES   WHOSE   LOWER    LIMIT 
WAS    ZERO,    AND    WHOSE    UPPER   LIMIT    WAS    NEAR   THE    ORIGINAL 
ELASTIC    LIMIT,    RUPTURE    DID     NOT    OCCUR    WITH    FROM    5    TO    1 6 

MILLION  REPETITIONS.  He  cautions  the  designer  (a)  that  this 
will  not  hold  for  DEFECTIVE  material,  i.e.,  a  factor  of  safety  must 
still  be  used  for  this  reason;  and  (b)  that  the  elastic  limit  of  the 
material  must  be  carefully  determined,  because  it  may  have  been 
artificially  raised  by  cold  working,  in  which  case  it  does  not  accur- 
ately represent  the  material.  The  original  elastic  limit  may  be  de- 
termined by  testing  a  piece  of  the  material  after  careful  annealing. 

(2)  WITH  OFTEN-REPEATED  STRESSES  VARYING  BETWEEN  ZERO 
AND  AN    UPPER  STRESS  WHICH    IS    IN  THE  NEIGHBORHOOD   OF   OR 
ABOVE  THE   ELASTIC  LIMIT,   THE   LATTER  IS   RAISED  EVEN  ABOVE, 
OFTEN  FAR  ABOVE,  THE  UPPER  LIMIT  OF  STRESS,  AND  IT  IS  RAISED 
HIGHER  AS  THE  NUMBER  OF   REPETITIONS   OF   STRESS  INCREASES, 

*  "  Ueber  die  Festigkeitsversuche  mit  Eisen  und  Stahl,"  A.  Wohler,  Berlin,  1870. 

f "  Mittheilungen  der  Konig.  Tech.  Hochschule  zu  Munchen,"  J.  Bau- 
schinger,  Munich,  1886  and  1897. 

J  "Structures  of  Iron  and  Steel,"  by  J.  Weyrauch.  Trans,  by  A.  J.  DuBoie, 
New  York,  1877. 


84  MACHINE  DESIGN. 

WITHOUT,    HOWEVER,    A    KNOWN    LIMITING    VALUE,    L,  BEING    EX- 
CEEDED. 

(3)  REPEATED  STRESSES  BETWEEN  ZERO  AND  AN  UPPER  LIMIT 
BELOW  L  DO  NOT  CAUSE  RUPTURE;  BUT  IF  THE  UPPER  LIMIT  IS 
ABOVE  L  RUPTURE  WILL  OCCUR  AFTER  A  LIMITING  NUMBER  OF 
REPETITIONS. 

From  this  it  will  be  seen  that  keeping  within  the  ORIGINAL 
ELASTIC  LIMIT  insures  safety  against  rupture  from  repeated 
stress  if  the  stress  is  not  reversed ;  and  that,  when  the  stress  is 
reversed,  the  total  range  should  not  exceed  the  ORIGINAL  ELAS- 
TIC RANGE  of  the  material. 

Various  formulae  have  been  proposed  by  different  authorities 
embodying  the  foregoing  laws. 

Unwin's  is  here  given  as  being  most  simple  and  general : 

Let  U  be  the  breaking  strength  of  the  material  in  pounds 
per  square  inch  for  a  load  once  gradually  applied. 

Let  /  max.  be  the  breaking  strength  in  pounds  per  square  inch 
for  the  same  material  subjected  to  a  variable  load  ranging  be- 
tween the  limits  /max.  and  /min.,  and  repeated  an  indefinitely 
great  number  of  times,  /min.  is  +  if  the  stress  is  of  the  same 
kind  as  /max.,  and  is  —  if  the  stress  is  of  the  opposite  kind, 
and  it  is  supposed  that  /  min.  is  not  greater  than  /  max.  Then 
the  range  of  stress  is  A—]  max.T/  min.,  the  upper  sign  being 
taken  if  the  stresses  are  of  the  same  kind  and  the  lower  if  they 
are  different.  Hence  A  is  always  positive.  The  formula  *  is 


/max.=  — 

where  ^  is  a  variable  coefficient  whose  value  has  been  experi- 
mentally determined.  For  ductile  iron  and  steel  r)  =  i.$,  in- 
creasing with  hardness  of  the  material  to  a  value  of  2. 

*  Unwin's  "Machine  Design,"  Vol.  i,  1903,  pp.  32-36. 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.    85 

This  formula  is  of  general  application. 
Three  cases  may  be  considered: 

(1)  A  constant  stress,  or  dead  load.     In  this  case  the  range 
of  stress,  J  =  o,  and  consequently  /max.  =  U,  as  it  should  be. 

(2)  The  stress  is  variable  between  an  upper  limit  and  zero, 
but  is  not  reversed. 

Here  4  =  f  max.,   since  /  min.  =o,  and  consequently  /  max.  = 


(3)  The  stress  is  reversed,   being  alternately  a  compressive 
and  tensile  stress  of  the  same  magnitude. 

Here  /min.  =  —  /max.     and     J  =  2/max. 


.*.  /max.  = — U. 

27] 


In  each  case  it  is  necessary  to  divide  the  breaking  load,  /  max., 
by  a  factor  of  safety  in  order  to  get  the  safe  unit  stress  /,  i.e., 

f  =  - — — — .    K  is  a  factor  of  safety  whose  numerical  value  depends 
K. 

upon  the  material  used.     (See  Sec.  51.) 

53.  Problem. — Consider  that  there  are  three  pieces  to  be 
designed  using  machinery  steel  having  an  ultimate  tensile  strength 
of  60,000  Ibs.  per  square  inch. 

The  first  piece  sustains  a  steady  load  having  a  dead  weight 
suspended  from  it. 

The  second  piece  is  a  member  of  a  structure  which  is  alter- 
nately loaded  and  unloaded  without  shock. 

The  third  piece  is  subjected  to  alternate  stresses  without 
shock. 

In  each  case  the  maximum  load  is  the  same,  being  30,000  Ibs. 
=F.  This  material  is  generally  reliable  and  uniform  in  quality. 
A  factor  of  safety  of  3  is  common;  /.  K  =  $  in  each  case;  9  =  1.5. 


86  MACHINE  DESIGN. 

CASE  I. 

/  max. 
/  max.  =  Z7    and     /  =  —     — . 

.     U     60000 
.*.  /  =  —  =    =  20,000  Ibs.  per  sq.  in. 

o  o 

Che  necessary  area  A  to  sustain  F  is  determined  by  the  equation 

F      30000       1 
A=-r= —   — =  if,s<7.  w. 

f          20000 

CASE  II. 

/  max.  =  2  (VV  +  i-  tj)  U. 

19  =  1.5     and     U  =  60,000  Ibs.  per  sq.  in. 
.*.  /  max.  =.605427  =  36,324  Ibs.  per  sq.  in 

/  max. 
7  =  —     —  =  12,108  Ibs.  per  sq.  in. 


F    30000  . 

A  =-7  = o  =2i  5<7-  *w-i  nearly. 

f     12108       2    *  7 


CASE  III. 


/  max.  = — U. 

1  27) 


60000 
/.  /  max.  = =  20,000  Ibs.  per  sq.  in. 


/  =  —     -'  =  6667  Ibs.  per  sq.  in. 

o 

F     30000 


The  importance  of  considering  the  question  of  range  of  stress 
in  designing  is  brought  home  by  this  illustration.     Comparison 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.     87 

of  results  shows  that  WITH  THE  SAME  MAXIMUM  LOAD  in  each  case, 
the  second  piece  must  be  given  nearly  twice  the  area  of  the  first, 
while  the  third  must  be  three  times  as  great  in  area  as  the  first, 
the  only  difference  in  the  three  cases  being  the  range  of  stress. 

Table  A  (page  88),  of  allowable  working  stresses,  as  compiled 
by  Unwin,  is  introduced  for  convenience  of  reference.  Such 
tables  are  to  be  used  with  judgment  in  reference  to  the  particular 
conditions  in  each  case. 

54.  Shock. — Consideration  of  the  design  of  parts  subjected 
to  shocks  or  suddenly  applied  loads. 

(1)  A  load  is  applied  on  an  unstrained  member  in  a  single 
instant,  but  without  velocity. 

In  this  case,  if  the  stress  does  not  exceed  the  limit  of  elasticity 
of  the  material,  the  stress  produced  will  be  just  twice  that  pro- 
duced by  a  gradually  applied  load  of  the  same  magnitude.  If 
F=  maximum  total  load  as  before,  then  the  maximum  total 
stress  =  2F.  The  design  of  the  member  is  then  made  as  in  Case 
II  or  Case  III  of  the  preceding  section. 

(2)  A  load  is  applied  on  an  unstrained  member  in  a  single 
instant,  but  with  velocity. 

In  this  case  the  stress  on  the  member  will  exceed  that  due  to 
a  gradually  applied  load  of  the  same  magnitude  by  an  amount 
depending  on  the  energy  possessed  by  the  load  at  the  moment 
of  impingement. 

Assume  that  a  member  is  stressed  by  a  load  F  falling  through 
a  height  h.  The  unknown  area  of  the  member  =A,  and  the 
allowable  strain  (i.e.,  extension  or  compression)  =  A.  As  before, 
/=  allowable  unit  stress  (determined  by  the  question  of  range  by 
the  use  of  Un win's  formula). 

The  energy  of  the  falling  load  is  F(h+  X). 

The  work  done  in  straining  the  member  an  amount  A  with  a 

maximum  fiber  stress  /  is  —  AA,   provided  the  elastic  limit  is 


88 


MACHINE  DESIGN, 
TABLE  A. — ORDINARY  WORKING  STRESS 

CASE  A.    DEAD  LOAD  INDUCING  PERMANENT  STRESS 


Kin 

d  of  Stress 

Tension 

Compression 

Bending 

Shear 

Torsion 

Cast  iron  

4200 

I2OOO 

6000-  8000 

2300 

4000-  6000 

Wrought  iron: 
Wrought  bar  or  forged  . 
Wrought  plate  ||   
Wrought  plate  j_. 

15000 
15000 

I200O 

15000 

15000 

I200O 
IOOOO 

7500 

Mild  steel  

I3OOO-I70OO 

13000-17000 

13000—1  7000 

10000—13000 

8000—12000 

Cast  steel  

Steel  castings 

I7OOO-2IOOO 
800O—I2OOO 

17000-21000 
12000—16000 

17000-21000 

13000-17000 

I2OOO-  I600O 

Phosphor  bronze  

IOOOO 

7000 

Gun-metal  

4200 

Rolled  copper  

4000 

2400 

Brass  .  .  

3000 

CASE  B.    VARYING  LOAD.       STRESS  FROM  2ERO  TO  A  MAXIMUM 


Kind  of  Stress 


Tension 

Compression 

Bending 

Shear 

Torsion 

Cast  iron 

2800 

8000 

4000—   5300 

1500 

2600—  4000 

Bar  iron  
Plate  iron  ||    
Plate  iron  j_  .  . 

IOOOO 
IOOOO 

8000 

IOOOO 

IOOOO 

8000 
6500 

5000 

Mild  steel  
Cast  steel  

8600-11400 
11400-14000 

8600-11400 
i  1400—14000 

8600-11400 
i  1400—14000 

6500-  8600 
8600—1  1400 

5300-  8000 
8000—10600 

Steel  castings  
Phosphor  bronze  

5300-  8000 
6600 

8000-10600 

6600-  9400 

4700-  8000 
4600 

4700  -8000 

Gun-metal  

2800 

Rolled  copper  

2600 

1600 

Brass  

2OOO 

CASE  C.   VARYING  LOAD.      ALTERNATE  EQUAL  STRESSES  OF  OPPOSITE  SIGN 


Material 

Tension  and 
Compression 

Bending  and 
Bending 

Shear  and 
Shear 

Torsion  and 
Torsion 

Cast  iron  
Bar  iron 

1400 
5000 

2000-2700 
5000 

770 
4000 

1300-2000 
2500 

Mild  s'eel 

4300—5700 

Cast  steel 

5700—7000 

5700—7000 

4300—5700 

Steel  castings  
Gun-metal  '.  .  . 

2700-4000 
1400 

3300-4700 

2300-4000 

2300-4000 

PROPORTIONS  OF  MACHINE  PAR  TS  AS  DICTA  TED  B  Y  STRESS-      89 

not  exceeded.  Equating  these  values  of  energy  expended  and 
work  done  and  solving  for  A  gives  A  =  2p(h  +  X)  +fL  Resilience 
is  the  physical  quality  determining  shock-resisting  capacity. 

55.  Form  Dictated  by  Stress.    Tension. — Suppose  that  A  and 
B,  Fig.  48,  are  two  surfaces  in  a  machine  to  be  joined  by  a  member 

subjected  to  simple  tension.  What  is  the  proper 
form  for  the  member?  The  stress  in  all  sec- 
tions of  the  member  at  right  angles  to  the  line 
of  application,  A  B,  of  the  force  will  be  the  same. 
Therefore  the  areas  of  all  such  sections  should  be  equal;  hence 
the  outlines  of  the  members  should  be  straight  lines  parallel  to 
AB.  The  distance  of  the  material  from  the  axis  AB  has  no 
effect  on  its  ability  to  resist  tension.  Therefore  there  is  nothing 
in  the  character  of  the  stress  that  indicates  the  form  of  the 
cross-section  of  the  member.  The  form  most  cheaply  produced, 
both  in  the  rolling-mill  and  the  machine-shop,  is  the  cylindrical 
form.  Economy,  therefore,  dictates  the  circular  cross-section. 
After  the  required  area  necessary  for  safely  resisting  the  stress  is 
determined,  it  is  only  necessary  to  find  the  corresponding  diam- 
eter, and  it  will  be  the  diameter  of  all  sections  of  the  required 
member  if  they  are  made  circular.  Sometimes  in  order  to  get  a 
more  harmonious  design,  it  is  necessary  to  make  the  tension 
member  just  considered  of  rectangular  cross-section,  and  this  is 
allowable  although  it  almost  always  costs  more.  The  thin,  wide 
rectangular  section  should  be  avoided,  however,  because  of  the 
difficulty  of  insuring  a  uniform  distribution  of  stress.  A  unit 
stress  might  result  from  this  at  one  edge  greater  than  the  strength 
of  the  material,  and  the  piece  would  yield  by  tearing,  although 
the  AVERAGE  stress  might  not  have  exceeded  a  safe  value. 

56.  Compression. — If  the   stress   be   compression   instead   of 
tension,  the  same  considerations  dictate  its  form  as  long  as  it 
is  a  "short  block,"  i.e.,  as  long  as  the  ratio  of  length,  to  lateral 
dimensions  is  such  that  it  is  sure  to  yield  by  crushing  instead  of 


90  MACHINE   DESIGN. 

by  "  buckling."  A  short  block,  therefore,  should  have  its  longi- 
tudinal outlines  parallel  to  its  axis,  and  its  cross-section  may  be 
of  any  form  that  economy  or  appearance  may  dictate.  Care 
should  be  taken,  however,  that  the  least  lateral  dimension  of 
the  member  be  not  made  so  small  that  it  is  thereby  converted 
into  a  "long  column." 

If  the  ratio  of  longitudinal  to  lateral  dimensions  is  such  that 
the  member  becomes  a  "long  column,"  the  conditions  that  dic- 
tate the  form  are  changed,  because  it  would  yield  by  buckling  or 
flexure  instead  of  crushing.  The  strength  and  stiffness  of  a 
long  column  are  proportional  to  the  moment  of  inertia  of  the 
cross-section  about  a  gravity  axis  at  right  angles  to  the  plane 
in  which  the  flexure  occurs.  A  long  column  with  "fixed"  or 
"  rounded"  ends  has  a  tendency  to  yield  by  buckling  which  is 
equal  in  all  directions.  Therefore  the  moment  of  inertia  needs 
to  be  the  same  about  all  gravity  axes,  and  this  of  course  points  to 
a  circular  section.  Also  the  moment  of  inertia  should  be  as  large 
as  possible  for  a  given  weight  of  material,  and  this  points  to  the 
hollow  section.  The  disposition  of  the  metal  in  a  circular  hollow 
section  is  the  most  economical  one  for  long-column  machine 
members  with  fixed  or  rounded  ends.  This  form,  like  that  for 
tension,  may  be  changed  to  the  rectangular  hollow  section  if 
appearance  requires  such  change.  If  the  long-column  machine 
member  be  "pin  connected,"  the  tendency  to  buckle  is  greatest 
in  a  plane  through  the  line  of  direction  of  the  compressive  force 
and  at  right  angles  to  the  axis  of  the  pins.  The  moment  of  iner- 
tia of  the  cross-section  should  therefore  be  greatest  about  a  gravity 
axis  parallel  to  the  axis  of  the  pins.  Example,  a  steam-engine 
connecting-rod. 

57.  Flexure. — When  the  machine  member  is  subjected  to 
transverse  stress  the  best  form  of  cross-section  is  probably  the  I 
section,  a,  Fig.  49,  in  which  a  relatively  large  moment  of  inertia, 
wUh  economy  of  material,  is  obtained  by  putting  the  excess  of 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.  QI 

the  material  where  it  is  most  effective  to  resist  flexure,  i.e.,  at  the 
greatest  distance  from  the  given  gravity  axis.  Sometimes,  how- 
ever, if  the  I  section  has  to  be  produced  by  cutting  away  the 
material  at  e  and  d,  in  the  machine-shop,  instead  of  producing 
the  form  directly  in  the  rolls,  it  is  cheaper  to  use  the  solid  rect- 
angular section  c.  If  the  member  subjected  to  transverse  stress 
is  for  any  reason  made  of  cast  material,  as  is  often  the  case,  the 
form  b  is  frequently  preferable  for  the  following  reasons: 

(1)  The  best  material  is  almost  sure  to  be  in  the  thinnest  part 
of  a  casting,  and  therefore  in  this  case  at  /  and  g,  where  it  is  most 
effective  to  resist  flexure. 

(2)  The  pattern  for  the  form  b  is  more  cheaply  produced  and 
maintained  than  that  for   a.     The  hollow  box  section,    when 
permitted  by  considerations  of  construction  and  expense  is  still 
better. 

(3)  If  the  surface  is  left  without  finishing  from  the  mold,  any 
imperfections  due  to  the  foundry  work  are  more  easily  corrected 
in  b  than  in  a. 

Machine  members  subjected  to  transverse  stress,  which  con- 
tinually change  their  position  relatively  to  the  force  which  pro- 
duces the  flexure,  should  have  the  same  moment  of  inertia  about 
all  gravity  axes,  as,  for  instance,  rotating  shafts  that  are  strained 
transversely  by  the  force  due  to  the  weight  of  a  fly-wheel,  or  that 
due  to  the  tension  of  a  driving-belt.  The  best  form  of  cross- 
section  in  this  case  is  circular.  The  hollow  section  would  give 
the  greatest  economy  of  material,  but  hollow  members  are  ex- 
pensive to  produce  in  wrought  material,  such  as  is  almost  inva- 
riably used  for  shafts.  The  hollow  circular  section  is  meeting 
with  increasing  use,  especially  for  large  shafts,  on  account  of  the 
combined  lightness  and  strength. 

58.  Torsion. — Torsional  strength  and  stiffness  are  propor- 
tional to  the  polar  moment  of  inertia  of  the  cross-section  of  the 
member.  This  is  equal  to  the  sum  of  the  moments  of  inertia 
about  two  gravity  axes  at  right  angles  to  each  other.  The  forms 


92  MACHINE   DESIGN. 

in  Fig.  49  are  therefore  not  correct  forms  for  the  resistance  of 
torsion.  The  circular  solid  or  hollow  section,  or  the  rect- 
angular solid  or  hollow  section,  should  be  used. 

The  I  section,  Fig.  50,  is  a  correct  form  for  resisting  the  stress 
P,  applied  as  shown.  Suppose  the  web  C  to  be  divided  on  the 
line  CD,  and  the  parts  to  be  moved  out  so  that  they  occupy 
the  positions  shown  at  a  and  b.  The  form  thus  obtained  is 
called  a  "box  section."  By  making  this  change  the  moment 
of  inertia  about  ab  has  not  been  changed,  and  therefore  the 
new  form  is  just  as  effective  to  resist  flexure  due  to  the  force  P 
as  it  was  before  the  change.  The  box  section  is  better  able  to 


FIG.  50. 

resist  torsional  stress,  because  the  change  made  to  convert  tne 
I  section  into  the  box  section  has  increased  the  polar  moment 
of  inertia.  The  two  forms  are  equally  good  to  resist  tensile  and 
compressive  force  if  they  are  sections  of  short  blocks.  But  if 
they  are  both  sections  of  long  columns,  the  box  section  would  be 
preferable,  because  the  moments  of  inertia  would  be  more  nearly 
the  same  about  all  gravity  axes. 

59.  Machine  Frames. — The  framing  of  machines  almost  always 
sustains  combined  stresses,  and  if  the  combination  of  stresses 
include  torsion,  flexure  in  different  planes,  or  long-column  com- 
pression, the  box  section  is  the  best  form.  In  fact,  the  box  sec- 
tion is  by  far  the  best  form  for  the  resisting  of  stress  in  machine 
frames.  There  are  other  reasons,  too,  besides  the  resisting  of 
stress  that  favor  its  use.* 


*  See  Richard's  "Manual  of  Machine  Construction.' 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.    93 

(1)  Its  appearance  is  far  finer,  giving  an  idea  of  complete- 
ness that  is  always  wanting  in  the  ribbed  frames. 

(2)  The  faces  of  a  box  frame  are  always  available  for  the 
attachment  of  auxiliary  parts  without  interfering  with  the  per- 
fection of  the  design. 

(3)  The  strength  can  always  be  increased  by  decreasing  the 
size  of  the  core,  without  changing  the  external  appearance  of 
the   frame,   and  therefore  without  any  work   whatever   on  the 
pattern  itself. 

The  cost  of  patterns  for  the  two  forms  is  probably  not  very 
different,  the  pattern  itself  being  more  expensive  in  the  ribbed 
form,  and  the  necessary  core-boxes  adding  to  the  expense  in  the 
case  of  the  box  form.  The  expense  of  production  in  the  foundry, 
however,  is  greater  for  the  box  form  than  for  the  ribbed  form, 
because  core  work  is  more  expensive  than  "green-sand  "  work. 
The  balance  of  advantage  is  very  greatly  in  favor  of  box  forms, 
and  this  is  now  recognized  in  the  practice  of  the  best  designers 
of  machinery. 

To  illustrate  the  application  of  the  box  form  to  machine 
members,  let  the  table  of  a  planer  be  considered.  The  cross- 
section  is  almost  universally  of  the  form  shown  in  Fig.  51.  This 
is  evidently  a  form  that  would  yield  easily  to  a  force  tending  to 


FIG.  51. 

twist  it,  or  to  a  force  acting  in  a  vertical  plane  tending  to  bend  it. 
Such  forces  may  be  brought  upon  it  by  "strapping  down  work," 
or  by  the  support  of  heavy  pieces  upon  centers.  Thus  in  Fig.  52 
the  heavy  piece  E  is  supported  between  the  centers.  For  proper 
support  the  centers  need  to  be  screwed  in  with  a  considerable 
force.  This  causes  a  reaction  tending  to  separate  the  centers  and 
to  bend  the  table  between  C  and  D.  As  a  result  of  this  the  V's 


94 


MACHINE  DESIGN. 


on  the  table  no  longer  have  a  bearing  throughout  the  entire 
surface  of  the  guides  on  the  bed,  but  only  touch  near  the  ends, 
the  pressure  is  concentrated  upon  small  surfaces,  the  lubricant  is 
squeezed  out,  the  V's  and  guides  are  "cut,"  and  the  planer  is 


FIG.  52. 


FIG.  53. 


FIG.  54. 


rendered  incapable  of  doing  accurate  work.  If  a  table  were  made 
of  the  box  form  shown  in  Fig.  53,  with  partitions  at  intervals 
throughout  its  length,  it  would  be  far  more  capable  of  maintain- 
ing its  accuracy,  of  form  under  all  kinds  of  stress,  and  would  be 
more  satisfactory  for  the  purpose  for  which  it  is  designed.* 

The  bed  of  a  planer  is  usually  in  the  form  shown  in  section 
in  Fig.  54,  the  side  members  being  connected  by  "  cross  -girts  " 
at  intervals.  This  is  evidently  not  the  best  form  to  resist  flexure 
and  torsion,  and  a  planer-bed  may  be  subjected  to  both,  either 
by  reason  of  improper  support  or  because  of  changes  in  the  form 
of  foundation.  If  the  bed  were  of  box  section  with  cross  parti- 
tions, it  would  sustain  greater  stress  without  undue  yielding. 
Holes  could  be  left  in  the  top  and  bottom  to  admit  of  supporting 
the  core  in  the  mold,  to  serve  for  the  removal  of  the  core  sand, 
and  to  render  accessible  the  gearing  and  other  mechanism  inside 
of  the  bed. 


*  Professor  Sweet  has  designed  and  constructed  such  a  table  for  a  large  mill- 
ing-machine. 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS    95 

This  same  reasoning  applies  to  lathe-beds.  They  are  strained 
transversely  by  force  tending  to  separate  the  centers,  as  in  the 
case  of  "chucking";  torsionally  by  the  reaction  of  a  tool  cutting 
the  surface  of  a  piece  of  large  diameter;  and  both  torsion  and 
flexure  may  result,  as  in  the  case  of  the  planer-bed,  from  an 
improperly  designed  or  yielding  foundation.  The  box  form 
would  be  the  best  possible  form  for  a  lathe -bed;  some  diffi- 
culties in  adaptation,  however,  have  prevented  its  extended  use 
as  yet. 

These  examples  illustrate  principles  that  are  of  very  broad 
application  in  the  designing  of  machines. 

60.  Brackets. — Often  in  machines  there  is  a  part  that  pro- 
jects either  vertically  or  horizontally  and  sustains  a  transverse 


FIG.  55. 


stress;  it  is  a  cantilever,  in  fact.  If  only  transverse  stress  is  sus- 
tained, and  the  thickness  is  uniform,  the  outline  for  economy  of 
material  is  parabolic.  In  such  a  case,  however,  the  outline  curve 
of  the  member  should  start  from  the  point  of  application  of  the 
force,  and  not  from  the  extreme  end  of  the  member,  as  in  the 
latter  case  there  would  be  an  excess  of  material.  Thus  in  A, 
Fig.  55,  P  is  the  extreme  position  at  which  the  force  can  be  ap- 


96  MACHINE  DESIGN. 

plied.  The  parabolic  curve  a  is  drawn  from  the  point  of  appli- 
cation of  P.  The  end  of  the  member  is  supported  by  the  auxil- 
iary curve  c.  The  curve  b  drawn  from  the  end  gives  an  excess 
of  material.  The  curves  a  and  c  may  be  replaced  by  a  single 
continuous  curve  as  in  C,  or  a  tangent  may  be  drawn  to  a  at  its 
middle  point  as  in  B,  and  this  straight  line  used  for  the  outline, 
the  excess  of  material  being  slight  in  both  cases.  Most  of  the 
machine  members  of  this  kind,  however,  are  subjected  also  to 
other  stresses.  Thus  the  "  housings"  of  planers  have  to  resist 
torsion  and  side  flexure.  They  are  very  often  supported  by  two 
members  of  parabolic  outline;  and  to  insure  the  resistance  of  the 
torsion  and  side  flexure,  these  two  members  are  connected  at  their 
parabolic  edges  by  a  web  of  metal  that  really  converts  them  into 
a  box  form.  Machine  members  of  this  kind  may  also  be  sup- 
ported by  a  brace,  as  in  D.  The  brace  is  a  compression  member 
and  may  be  stiffened  against  buckling  by  a  "web  "  as  shown,  or 
by  an  auxiliary  brace. 

61.  Other  Considerations  Governing  Form. — One  considera- 
tion governing  the  form  of  machine  parts  has  been  touched  upon 
in  the  preceding  sections.  It  may  be  well  to  state  it  here  as  a 
general  principle:  Other  considerations  being  equal  the  form  of 
a  member  should  be  that  which  can  be  most  cheaply  produced 
both  as  regards  economy  of  material  and  labor. 

Another  element  enters  into  the  form  of  cast  members.  Cast- 
ings, unless  of  the  most  simple  form,  are  almost  invariably  sub- 
jected to  indeterminate  shrinkage  stresses.  Some  of  these  are 
undoubtedly  due  to  faulty  work  on  the  part  of  the  molder,  others 
are  induced  by  the  very  form  which  is  given  the  piece  by  the  de- 
signer. They  cannot  be  eliminated  entirely,  but  the  danger  can 
be  minimized  by  paying  attention  to  these  general  laws: 

(a)  Avoid  all  sharp  corners  and  re-entrant  angles. 

(b)  All  parts  of  all  cross-sections  of  the  member  should  be 
as  nearly  of  the  same  thickness  as  possible. 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.  97 

(c)  If  it  is  necessary  to  have  thick  and  thin  parts  in  the  same 
casting,  the  change  of  form  from  one  to  the  other  should  be  as 
gradual  as  possible. 

(d)  Castings  should  be  made  as  thin  as  is  consistent  with  con- 
siderations of  strength,  stiffness,  and  resistance  to  vibration. 

The  following  tables  of  properties  of  materials,  and  formulae 
of  mechanics  are  grouped  here  for  convenient  reference.  Tables 
of  strength  of  materials  must  be  understood  to  represent  ap- 
proximate average  results.  This  property  varies  not  only  with 
the  chemical  constitution  but  also  with  the  physical  condition 
as  affected  by  heat  treatment,  jK).t  or  cold-working,  etc.,  and 
even  by  the  size  and  form  of  the  part.  It  is  impossible  to  tabulate 
with  reference  to  these  factors.  The  machine  designer  must 
have  an  exhaustive  knowledge  of  the  properties  of  materials 
of  construction  and  of  the  factors  affecting  these  properties. 
Only  such  knowledge  can  guide  him  in  the  selection,  say,  of 
wrought  iron  rather  than  mild  steel  for  a  piece  whose  subsequent 
strength  might  be  imperiled  by  overheating  in  the  course  of  its 
manufacture;  or,  again,  in  the  selection  of  a  suitable  tough, 
shock-resisting,  alloy  steel  for  a  piece  destined  for  severe  service, 
which  might  readily  fail  if  made  of  a  harder,  less  yielding,  bat 
apparently  "  stronger  "  material. 


t 


98  MACHINE  DESIGN. 

TABLE  B. — PHYSICAL  PROPERTIES  OF  METALS. 


Metal 

Melting- 
point 
Degrees 
Fahr. 

Specific 
gravity 
at  ordin- 
ary temps 

Struct- 
ure* 

Electric  con- 
ductivity. 
Silver  100  at 
32°  F. 

Approx.  value 
per  pound, 
Dollars  (1915) 

Weight 
per    cubic 
inch, 
Pounds 

Aluminum.  .  . 

1214. 

2   6^ 

M 

CQ    2 

O    IO 

OOs6 

Antimony.  .  . 

1166 

6    71 

B 

4    J 

O    I  ^ 

2421 

Barium  

1^62 

7     7C 

M 

2800  oo 

13^3 

Bismuth  

516 

0.80 

B 

I  .  I 

2    8O 

3^36 

Cadmium  
Calcium  
Chromium.    . 

610 
1436 
2712 

8.64 

i-55 
6  c,o 

M 
M 
B 

24.2 
21.3 

i-35 
3.00 

I     C.O 

.3118 
•0559 

2^46 

Cobalt  

2714 

8  60 

M 

16  3 

4   OO 

^IO^ 

Copper  

1082 

8.85 

M 

Q2  .  2 

O    14 

7IQA 

Gold  

1944 

IQ  .  ^2 

M 

69.8 

• 

3OO    OOQ 

.6O72 

Iron,  cast  
Iron,  pure  
Iron,  wrought.  .  .  . 
Lead  

220of 
2975 

28oof 

621 

7.20 
7.86 
7.70 

II    37 

B 

M 
M 

s 

1.9 
15.0 

14.8 

7  6 

O.OI 

O.O2 
O   O4 

.2598 
.2836 
.2779 
4103 

M  agne  si  um 

1172 

I    74 

M 

•2  r    2 

2    C.O 

0628 

Manganese  

22O5t 

8.00 

B 

I     C.O 

.2887 

Mercury 

—  37   8 

I  3    60 

F 

I    6 

O    7^ 

4008 

Nickel  
Platinum  
Potassium  
Silver  
Steel,  machinery. 

Steel  tool 

2647 
3190 
144 
1760 
257°t 

2C,7ot 

8.80 

21  .50 

0.86 
10.50 

7-7 

7   8<? 

M 
M 

S 
M 
M 

M 

II.7 
15-6 

6.4 

IOO.O 
IO.O 

[      3-4t 
\       to 

0.50 
SOO.OO 

16.00 
8.00 

O.O2 

0.06  ) 

to       > 

•3175 
.7758 
•0314 
.3789 
.278 

2833 

Tin 

440 

7    20 

M 

I    10.  ot 
II    I 

LU                f 

1  .00    J 

O    34 

262,1 

Tungsten 

CC44 

17   60 

B 

33    O 

I    7^ 

63^1 

Vanadium       .    . 

2047 

e    ro 

M 

104  oo 

.108? 

Zinc    

784 

7  .  IO 

M 

27  .0 

0.06 

.  2^62 

*  B  =  Brittle,  F  =  Fluid,  M  =  Malleable,  S  =Soft. 

1  Varies.     Approx.  Av.  Value  Only. 
Glass  hard,  3-4-     Soft,  10.00. 
$20.67  per  ounce  troy. 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.    99 


o  o  o       o  o 


«   4->  •*-» 

"s.5-1-  g 

Jiff 

—    i-         O 


00   t^ 
10  10 

co  CO 


ills 

o  bo  ,0 


10        O         M         CN 


«  o. 
£3 
3*8 
•3  « 

II 
11 


•3.3 


22 


.•8>. 

|||  4 

t2  oj2    ^" 


O  O  O  O  «0  10 
O"  O"  o"  10  ^o'  co 
10  t-^  <N  co 


w  o   :  2s 

MM  M 


£  6CJ3  „ 

g.Sti  d 

Isg  •< 

-J3  «  £  n 


88 


C  en         H   <u 

oc.fi     55 

.fa    O—    O         ^  to 


if 


U       «3       prt 


O 
| 

s:l 


I  • 

|    V 

ll 

bC  G 


100 


MACHINE  DESIGN. 


TABLE  D. — ELEMENTS  or  USUAL  SECTIONS. 

Moments  refer  to  horizontal  gravity  axis,  as  shown.  Values  for  flanged  beams 
apply  to  standard  minimum  sections  only.  A  =  area  of  section,  c  =  distance 
neutral  axis  to  outer  fiber. 


Shape  of 
Section. 


Moment  of 
Inertia,  7. 


Section  Modulus, 


Distance  of 

Base  from  Center 

of  Gravity. 


Least  Radius 
of  Gyration. 


bh*_ 

12 


bh* 
6 


Least  side 


36 


24 


The  lesser  of 

h  b 

-  or  - 

4.24        4.9 


64 


32 


a   dS-dS 
0.0982- — — 
"i 


o.iogSr4 


o.iQoSr 


64 


7r6a! 
32 


PROPORTIONS  OF  MACHINE  PARTS  AS  DICTATED  BY  STRESS.    101 


TABLE  D. — ELEMENTS  OF  USUAL  SECTIONS. — Continued. 

Approximate. 


Shape  of 
Section. 

Moment  of 
Inertia,   /. 

Section  Modulus, 
/ 

e 

Distance  of 
Base  from  Center 
of  Gravity. 

Least  Radius 
of  Gyration. 

•y 

Ah* 
10.4 

Ah 

7-4 

h 
3-5 

k 

5 

ii 

1 

I 

Ah* 
6.1 

Ah 
3 

h 

2 

_b_ 

5-2 

_-.  i  — 

I 

Ah2 
6.73 

Ah 
3-3 

ft 

2 

b 
3-56 

Polar  Moment  of  Inertia  J. 

Polar  Modulus  of  Section  —  . 

'• 

bh*+hb* 

i    bh*+hb3 

12 

6    V6*+A* 

}• 

&!4-M 

I      &!4-&24 

5 

6 

3      V^ 

i 

0 

yd* 

32 

TrJ3 
16 

®\ 

0.0982(6 

74      fj  4"> 

^    rfl4-rf24 

n         U2  ) 

•  I9°4       j 
«i 

01 

4—b~+ 

i(&a'+a&3) 

TT  /6a3+a63\ 

3^1       a       j 

102 


MACHINE  DESIGN. 
TABLE  E. — BEAMS  OF  UNIFORM  CROSS-SECTION 


Maximum 
moment 

Maximum 
deflection 

Cantilever  single  load  at  end 

Wl 

i  Wl* 

Cantilever,  uniform  load  

\  Wl 

3  El 

i  Wl3 

Simple  beam  load  at  middle            .    . 

i  Wl 

8  El 

i   Wl3 

Simple  beam  uniformly  loaded 

i  Wl 

48  El 
5    Wl3 

Beam  fixed  at  one  end,  supported  at  other,  load  near 
middle                                     .          

o  192  Wl 

384  El 

o  Wl3 
o  0008  —  — 

Beam  fixed  at  one  end,  supported  at  other,  uniform 
load                                           

1  Wl 

^    El 

Wl3 
o  .  0054  -=7 

Beam  fixed  at  both  ends  load  at  middle 

i  Wl 

El 
i    Wl3 

Beam  fixed  at  both  ends  uniform  load    

I/I2   Wl 

192  El 
i    Wl3 

384  El 

W=  total  load,  1  =  length,  E  =  Young's  modulus,  /  =  moment  of  inertia. 
See  further  any  handbook,  or  treatise  on  strength  of  materials. 


TABLE  F. — STRESS  AND  STRAIN  FORMULAE 


Unit  stress  in  tension  or  compression  

/.«/«=£ 

Strain  in  tension  or  compression      

x-4 

AE 

Unit  stress  in  shear 

/•-- 

Js    A 

Unit  torsional  stress 

fs=M*     M-Pa 

,      i6Mt 

ft     <*d* 

Torsional  strain 

0_i8oMtl 

0         JG 

Unit  stress  in  flexure    

Mbc 

Jb        j 

Deflection  in  flexure 

See  Table  E 

Unit  stress,  combined  flexure  and  ten.  or  comp. 

,    P  .  Mbc 
f=  T+"T~     (straight  axis) 
•ft-       i                                 \ 

PROPOR  TIONS  OF  MACHINE  PAR  TS  AS  DICTA  TED  BY  S  TRESS.     103 
TABLE  F. — STRESS  AND  STRAIN  FORMULA — Continued. 


Combined  torsion  and  flexure  

Meb=  O.3$Mb-\-O.6$vMb2-\-Mt2 

Ditto,  unit  resultant  normal  tensile  stress;   cir. 
shaft  . 

fl      2(Mb~\-  v  Mtz-\~Mb2 

'                        irr* 

Ditto,  unit  resultant  diagonal  shear;  cir.  shaft.  . 

.,  _  2\/Mt2+Mb2 

J   *                     TIT3 

Combined  tension  and  torsion  

/=  0.35/1+0.65  V/,2+4/s2 

Combined  compression  and  torsion,  long  column 

7T2         P            Mt2 

I2     El  '  4£2/2 

Long  Columns  

k=  factor  of  safety 

One  end  free  other  fixed 

P    ir2EI  mm 

k            4/2 

Both  ends  free,  guided  in  direction  of  load  

P    ir2EI  mm 

k            I2 

One  end  fixed,  other  free  but  guided  in  direction 
of  load 

P     27r2E7  min 

k             I2 

Both  ends  fixed  in  direction 

P     4-ir-EI  mm 

k             I2 

P  =  force,    A  =  area,    /  =  length,    M&  =  bending   moment,    M t  =  torsional     moment,  —  = 

section  modulus,   flexure, —  =  section  modulus,  torsion,    E  =  modulus  of  elasticity,    G- 
modulus  of  shear. 

TABLE  G. — STRESSES  IN  PRESSURE  VESSEL  WALLS 

=  unit  excess  internal  pressure,  Ibs.  per  sq.  in.);   (1  =  thickness  of  plate,  ins.). 


Thin  cylinder,  stress  in  long,  section  

^••- 

(Z>2=  inner  diam.,  in.) 

Thin  cylinder,  stress  in  cir.  section  

f_Djp 

4/ 

Thin  sphere,  stress  in  any  section  .... 

,_D-2p 

4/ 

. 

Ihick  cylinder,  stress  in  long.  sec.     rree 

D      tfJIO/+ 

jp       Birnie,  and 

theory)  

i3/>                 Grashof. 

Ditto,  fixed  or  solid  ends.     (Max.  strain 
theory.)  

^=W^r 

i3/>             Claverino. 

Ditto 

t    0.42  pD-2 

1   f-p 

Sames. 

Ditto,  free  ends.     (Max.  stress  theory.)  .... 

*      at 

Barlow. 

Ditto,  either  free  or  fixed  ends.     (Guest's 
max.  shear  law.)  

J=2Dl\L~D 

")                        Moss. 

104  MACHINE  DESIGN. 

TABLE  G. — STRESSES  IN  PRESSURE  VESSEL  WALLS. — Continue 
Flat  Plates,  Uniformly  loaded. 


Uniformly    stayed,    rows    a   inches    apart 
/&=  unit  flexural  stress  

f»=*~p 

Integral  cast-iron  cyl.  head.  R%=  inner  rad. 
r=  rad.  of  inner  curv.  of  flange 

/A-o.^H-iU', 

Bach. 

Flanged  steel,  riveted  cyl.  head.     R2  and  r 

/»> 

Bach. 

Flat  circular  unstayed  plate,  not  flanged. 
0=  A  to  f  for  cast-iron  heads 
=  f  to  §  for  free-lying  steel  heads 
=  ^  to  ^  for  st.  heads,  rigidly  fastened  at 
circumference 
=  f  to  ^  for  st.  heads  fastened  at  cir.  but 
yielding  to  equalize  stress  at  center 
and  cir. 

Bach. 

Elliptical  plate;    a,  major,  b,  minor  axis. 
<h  —  4  to  "§  cast  iron 

/              2               p 

^    I+\aj                      Bach. 

Rectangular  plate;   a,  major,  b,  minor  side. 
<£=  i  to  If  cast  iron       

Same  as  foregoing. 

Square  plate;  side=  a.     </>=  f  to  f,  cast  iron 

~  2  \    /&                                   Bach. 

Concentrated  Load  at  Center=P. 
(Plates  supported,  but  not  fixed,  at  edges). 

Circular  plate,  rad.  of  load=^0.    <£=f,  cast 
iron 

TT     \       3  1?  //ft                     Bach. 

Rectangular  plate,  a,  major,  6,  minor  side.  . 
0=  -j  to  2   cast  iron  

,>      /            i       P 

\       £.+*'/* 

6  ^a                          Bach. 

Elliptical  plate,  o,  major,  b,  minor  axis. 
<6  —  -f  to  If  cast  iron 

/  0     8+4  1  —  )   +3(-)      ,    „ 
.  >      /  o             \a  /          \  a  /      o  P 

~  \/  ^^^  j_    fi  \  2_i_    /^  \  4   a  /6 
Bach. 

CHAPTER  VII. 


RIVETED    JOINTS. 

62.  Methods  of  Riveting. — A  rivet  is  a  fastening  used  to  unite 
metal  plates  or  rolled  structural  forms,  as  in  boilers,  tanks,  built- 
up  machine  frames,  etc.  It  consists  of  a  head,  .4,  Fig.  56,  and  a 
straight  shank,  B.  It  is  inserted,  usually  red-hot,  into  holes, 
either  drilled  or  punched  in  the  parts  to  be  connected,  and  the 
projecting  end  of  the  shank  is  then  formed  into  a  head  (see  dotted 
lines)  either  by  hand-  or  machine-riveting.  A  rivet  is  a  permanent 
fastening  and  can  only  be  removed  by  cutting  off  the  head.  A 
row  of  rivets  joining  two  members  is  called  a  RIVETED  JOINT  or 
SEAM  OF  RIVETS.  In  hand-riveting  the  projecting  end  of  the 
shank  is  struck  a  quick  succession  of  blows  with  hand  hammers 
and  formed  into  a  head  by  the  workman.  A  helper  holds  a  sledge 


W 
FIG.  57. 


FIG.  58. 


or  "dolly  bar"  against  the  head  of  the  rivet.  In  "  button  -set " 
or  "snap  "  riveting,  the  rivet  is  struck  a  few  heavy  blows  with  a 
sledge  to  "upset"  it.  Then  a  die  or  "button  set,"  Fig.  57,  is 
held  with  the  spherical  depression,  B,  upon  the  rivet;  the  head, 
A,  is  struck  with  the  sledge,  and  the  rivet  head  is  thus  formed. 
In  machine-riveting  a  die  similar  to  B  is  held  firmly  in  the  ma- 
chine and  a  similar  die  opposite  to  it  is  attached  to  the  piston  of 


106  MACHINE  DESIGN. 

a  steam,  hydraulic,  or  pneumatic  cylinder.  A  rivet,  properly 
placed  in  holes  in  the  members  to  be  connected,  is  put  between 
the  dies  and  pressure  is  applied  to  the  piston.  The  movable  die 
is  forced  forward  and  a  head  formed  on  the  rivet. 

The  relative  merits  of  machine-  and  hand-riveting  have  been 
much  discussed.  Either  method  carefully  carried  out  will  pro- 
duce a  good  serviceable  joint.  If  in  hand-riveting  the  first  few 
blows  be  light  the  rivet  will  not  be  properly  upset,  the  shank  will 
be  loose  in  the  hole,  and  a  leaky  rivet  results.  If  in  machine- 
riveting  the  axis  of  the  rivet  does  not  coincide  with  the  axis  of 
the  dies,  an  off -set  head  results.  (See  Fig.  58.)  In  large  shops 
where  work  must  be  turned  out  economically  in  large  quantities, 
machines  must  be  used.  But  there  are  always  places  inacces- 
sible to  machines,  where  the  rivets  must  be  driven  by  hand.* 

63.  Perforation  of  Plates. — Holes  for  the  reception  of  rivets 
are  usually  punched,  although  for  thick  plates  and  very  careful 
work  they  are  sometimes  drilled.  If  a  row  of  holes  be  punched 
in  a  plate,  and  a  similar  row  as  to  size  and  spacing  be  drilled  in 
the  same  plate,  testing  to  rupture  will  show  that  the  punched  plate 
is  weaker  than  the  drilled  one.  If  the  punched  plate  had  been 
annealed  it  would  have  been  nearly  restored  to  the  strength  of 
the  drilled  one.  If  the  holes  had  been  punched  ^  inch  to  J  inch 
small  in  diameter  and  reamed  to  size,  the  plate  would  have  been 
as  strong  as  the  drilled  one.  These  facts,  which  have  been  ex- 
perimentally determined,  point  to  the  following  conclusions: 
First,  punching  injures  the  material  and  produces  weakness. 
Second,  the  injury  is  due  to  stresses  caused  by  the  severe  action 
of  the  punch,  since  annealing,  which  furnishes  opportunity  for 
equalizing  of  stress,  restores  the  strength.  Third,  the  injury 
is  only  in  the  immediate  vicinity  of  the  punched  hole,  since  ream- 
ing out  ^  inch  or  less  on  a  side  removes  all  the  injured  material. 

*  See  Sec.  75  for  discussion  of  the  importance  of  holding  rivet  under  pressure 
until  it  is  cooled;  and  the  advantage  of  large  rivets  over  small. 


RIYEJED  JOINTS. 


107 


In  ordinary  boiler  work  the  plates  are  simply  punched  and  riveted. 
If  better  work  is  required  the  plates  must  be  drilled,  or  punched 
small  and  reamed,  or  punched  and  annealed.  Drilling  is  slow 
and  therefore  expensive;  annealing  is  apt  to  change  the  plates 
and  requires  large  expensive  furnaces.  Punching  small  and  ream- 
ing is  probably  the  best  method.  In  this  connection,  Prof.  A. 
B.  W.  Kennedy  (see  Proc.  Inst.  M.  E.,  1888,  pp.  546-547)  has 
called  attention  to  the  phenomenon  of  greater  unit  tensile  strength 
of  the  plate  along  the  perforations  than  of  the  original  unperforated 
plate.*  Stoney  (".Strength  and  Proportion  of  Riveted  Joints," 
London,  1885)  has  compiled  the  following  table: 

TABLE  I. — RELATIVE  PERCENTAGE  OF  STRENGTH  OF  STEEL  PLATES  PERFORATED 
IN  DIFFERENT  WAYS. 


Specimens. 

Unit  Strength  of  Net  Section  between  Holes  compared 
with  that  of  the  Solid  Plate  (100  Per  Cent). 

i  Inch. 

*  Inch. 

i  Inch. 

i  Inch, 

Punched  

Per  Cent. 
101  .0 
105.6 
113.8 

Per  Cent 
94.2 
105.6 
III.  I 

Per  Cent. 
82.5 

IOI.O 

106.4 

Per  Cent. 
75-8 
100.3 
106.  i 

Punched  and  annealed  

Drilled  

For  punched  and  reamed  holes  the  same  percentages  may 
be  used  as  for  drilled. 

Professor  Kennedy  gives  constants  which  may  be  obtained 
from  the  following  formula:  Excess  of  unit  strength  of  drilled 
steel  plates  in  net  section  over  unperforated  section 


( 


/  is  the  thickness  of  plate  in  inches  and  r  the  ratio  of  pitch 
divided  by  diameter  of  hole.  No  data  exist  relative  to  iron 
plates  in  this  matter.  If  7  =  4.5,  or  more>  there  is  no  excess. 

64.  Kinds  of  Joints.  —  Riveted  joints  are  of  two  general  kinds: 
First,  LAP-JOINTS,  in  which  the  sheets  to  be  joined  are  lapped  on 

"This  reported  phenomenon  is  corroborated  by  tests  made  at  Watertown 
Arsenal.  See  Tests  of  Metals,  1886,  pp.  1264,  1557.  It  is  fully  explained  by  tht 
condition  of  localized  stress  and  the  consequent  prevention  of  lateral  contraction. 


io8 


MACHINE  DESIGN. 


each  other  and  joined  by  a  seam  of  rivets,  as  in  Fig.  590.  Second, 
BUTT-JOINTS,  in  which  the  edges  of  the  sheets  abut  against  each 
other,  and  a  strip  called  a  "  cover-plate "  or  "  butt-strap"  is  riv- 
eted to  the  edge  of  each  sheet,  as  in  c.  In  recent  years  a  lap-joint 


oii 


OQj 

oo! 

L 

op 
ob 

1 

Oii    ( 

0    !! 

OH 

o  o 

0   0 

©©©©©© 

._.§__©._©_©___©.J 
""©"©"©""©"©"© 
©   ©   ©  ©   © 


^ 


FIG.  60. 

with  a  single  cover-plate  has  been  used  somewhat.     It  is  shown 
in  Fig.  S9e- 

There  are  two  chief  kinds  of  riveting:  Single,  in  which  there 
is  but  one  row  of  rivets,  as  in  Fig.  59*2;  and  double,  where  there 
are  two  rows. 


RIVETED  JOINTS. 


109 


Double  riveting  is  subdivided  into  "  chain-riveting,"  Fig.  596, 
and  "  zigzag  "  or  "  staggered  "  riveting,  Fig.  59  d. 

Lap-joints  may  be  single,  double  chain,  or  double  staggered 
riveted. 

Butt-joints  may  have  a  single  strap  as  in  c,  or  double  strap; 
i.e.,  an  exactly  similar  one  is  placed  on  the  other  side  of  the  joint. 
Butt-joints  with  either  single  or  double  strap  may  be  single, 


©i 


Si 

©: 


© 
© 


©©©©©©©© 
©©_©©_©©© 

©~~~©~~©""©~©~"©"©~~© 
©©©©©©©© 


FIG.  6oA. 

double  chain,  or  double  staggered  riveted.     In  butt-joints,  single- 
cover-plates  should  have  a  thickness  =  /  + J" ;    and  double  cover- 
plates  =  — H —  ";  /  being  the  thickness  of  the  main  plates. 
To  sum  up,  there  are: 


Lap-joints. 


Butt-joints. 


f  Single-riveted 

•j  Double  chain-riveted 

[  Double  staggered-riveted 

f  Single-riveted 

Single-strap j  Double  chain-riveted 

I  [  Double  staggered-riveted 

f  Single-riveted 

i  Double-strap -I  Double  chain-riveted 

[  [  Double  staggered-riveted 


The  demands  of  modern  practice  have  added  triple,  quad- 
ruple and  quintuple  joints  to  the  foregoing.     In  high-pressure 


no 


MACHINE  DESIGN. 


cylindrical  boilers,  for  instance,  common  practice  is  to  employ 
for  the  longitudinal  seam  the  highly  efficient  joint  shown  in 
Fig.  60.  Here  we  have  a  triple-riveted  butt-joint  with  double 
cover-plates;  on  each  side  of  the  joint  two  rows  of  rivets  are  in 


© 

© 

^'"'"© 

© 

(9) 

.-'"'""© 

© 

© 

© 

r'~©       ©        © 

© 

© 

© 

© 

T^f 

T©  ©  ©  ©  ©  © 

©  © 

©  © 

©      © 

©     | 

[©©©©© 

©  © 

©  © 

©      © 

©      © 

T  ©  ©  ©  ©  © 

©  © 

©  © 

©      © 

©      1 

v^) 

J©  ©  ©  ©  ©  © 

©  © 

©  © 

©      © 

©     j 

a 

k©       ©       © 

© 

© 

© 

© 

© 
© 

"^^-^ 

© 

© 

/JE\ 
\27 

© 

- 

~] 

(0) 

i< 

Pitch 

'' 

FIG.  6oB. 

double  shear  and  one  row,  the  outer,  is  in  single  shear. 
Quadruple  and  quintuple  joints  are  shown  in  Figs.  6oA  and  6oB. 
65.  Failure  of  Joint. — A  riveted  joint  may  yield  in  any  one  of 
four  ways:  First,  by  the  rivet  shearing  (Fig.  61  a);  second,  by  the 
plate  yielding  to  tension  on  the  line  AB  (Fig.  61  b) ;  third,  by  the 
rivet  tearing  out  through  the  margin,  as  in  c;  fourth,  the  rivet 
and  sheet  bear  upon  each  other  at  D  and  E  in  d,  and  are  both 
in  compression.  If  the  unit  stress  upon  these  surfaces  becomes 
too  great,  the  rivet  is  weakened  to  resist  shearing,  or  the  plate 
to  resist  tension,  and  failure  may  occur.  This  pressure  of  the 


FIG.  61. 


RIVETED  JOINTS.  Ill 

rivet  on  the  sheet  is  called  "bearing  pressure."  It  is  obvious 
that  the  strongest  or  most  efficient  joint  in  any  case  will  be  one 
which  is  so  proportioned  that  the  tendency  to  fail  will  be  equal 
in  all  of  the  ways. 

66.  Strength  of  Materials  Used. — As  a  preliminary  to  the 
designing  of  joints  it  is  necessary  to  know  the  strength  of  the 
rivets  to  resist  shear,  of  the  plate  to  resist  tension,  and  of  the 
rivets  and  plates  to  resist  bearing  pressure.  These  values  must 
not  be  taken  from  tables  of  the  strength  of  the  materials  of  which 
the  plate  and  rivets  are  made,  but  must  be  derived  from  experi- 
ments upon  actual  riveted  joints  tested  to  rupture.  The  reason 
for  this  is  that  the  conditions  of  stress  are  modified  somewhat  in 
the  joint.  For  instance,  in  single-strap  butt-joints,  and  in  lap- 
joints,  the  line  of  stress  being  the  center  line  of  plates,  and  the 
plates  joined  being  offset,  flexure  results  and  the  plate  is  weaker 
to  resist  tension,  the  rivets  in  the  mean  time  being  subjected  to 
tension  as  well  as  shear;  if  the  joint  yield  to  this  stress  in  the 
slightest  degree  the  "bearing  pressure  "  is  localized  and  becomes 
more  destructive.  The  effect  of  friction  between  the  surfaces 
of  the  plates  under  the  pressure  at  which  they  are  "gripped  " 
by  the  rivets  is  another  item  of  considerable  importance.  Ex- 
tensive and  accurate'  experiments  have  been  made  upon  actual 
joints  and  the  results  have  been  published.* 

The  table  on  page  112  has  been  compiled  as  representing  fair 
average  results,  and  the  values  there  given  may  be  used  for  ordi- 
nary joints. 

The  Master  Steam  Boiler-Makers'  Assn.,  as  the  result  of 
tests  conducted  by  its  committee,  recommends,  for  iron  rivets, 
fs=  42000  and  fs  =  40000 ;  for  steel  ri  vets,  /«  =  46000,  f's  =44000. 

It  will   be  noted  that  the  values  of  ft  are  not  given  for  steel. 

*  See  Proc.  Inst.  of  Mech.  Eng.,  1881,  1882,  1885,  1888;  Tests  of  Metals, 
Watertown  Arsenal,  1885,  1886,  1887,  1891,  1895,  1896.  Stoney's  "Strength  and 
Proportions  of  Riveted  Joints,"  London;  1885. 


112 


MACHINE  DESIGN. 


TABLE  II. — VALUES  OF  ft,  /«,  AND  /c  FOR  DIFFERENT  KINDS  OF  JOINTS. 


Iron. 

Steel. 

Kind  of  Joint. 

ft 

fs 

fc 

ft 

f* 

fc 

Lap-joint,    single-  riveted,     punched 
holes  

40000 

38000 

67000 

47^oo 

85000 

Lap-joint,  single-riveted,  drilled  holes 

45000 

36000 

67000 

45000 

85000 

Lap-joint,    double-riveted,    punched 
holes 

1  CQOO 

40000 

67000 

48000 

Lap-joint  double-riveted  drilled  holev 

^OOOO 

38000 

67000 

46000 

85000 

Butt-joints,  single  cover  :    Use  values 
given  for  lap-joints. 
Butt-joints,  double  cover,  single-riv- 
eted  punched  holes 

4OOOO 

/•' 

fc' 
80000 

/•' 

48000 

fc' 

Butt-joints,  double  cover,  single-riv- 
eted, drilled  holes  
Butt-joints,  double  cover,  double-riv- 
eted  punched  holes     

45000 

4rooo 

41000 
38000 

89000 

80000 

46000 
47<OO 

I  00000 
I  OOOOO 

Butt-joints,  double  cover,  double-riv- 
eted  drilled  holes 

50000 

36000 

80000 

4.^000 

IOOOOO 

*  Original  plate 

50000 

60000 

*  Original  bar 

4.COOO 

52000 

The  tensile  strength  of  steel  varies  through  a  considerable  range 
due  largely  to  differences  in  chemical  constitution ;  it  also  follows 
a  rough  law  of  inverse  proportion  to  the  thickness  of  plates; 
i.e.,  thin  plates  will  be  almost  sure  to  show  higher  tensile  strength 
than  thicker  plates  of  the  same  composition.  Furthermore,  the 
method  of  perforation  greatly  affects  the  strength  of  the  plates, 
as  has  been  pointed  out  in  §  63.  Ordinary  boiler  plates  have  a 
unit  tensile  strength  ranging  from  55,000  Ibs.  to  62,000  Ibs.  per 
square  inch.  For  ordinary  calculations  ft  may  be  taken  as 
55,000  Ibs.  for  punched  plates  and  60,000  for  drilled  plates. 
The  shearing  strength  of  rivets  also  varies  inversely  as  their  size, 
but  these  differences  are  slight. 

The  Boiler  Code  Committee  of  the  A.  S.  M.  E.  recommend: 
for   iron    rivets,   /,  =  38,000    and  f's  =  35,000;    for    steel    rivets, 


*  If  the  original  material  varies  from  this,  the  values  given  above  should  be 
varied  proportionately. 


RIVETED  JOINTS.  113 

fs  =  42,000  and  /',  =  39,000.  They  also  recommend  as  the  max- 
imum values  to  be  used :  for  iron  rivets,  fs  =  38,000  and  /',  =  38,000 ; 
for  steel  rivets,  fs  =  44,000  and  /',  =  44,000.  They  recommend 
ft  =  55,000  for  mild  steel,  and  ft  =  45,000  for  wrought  iron,  where 
the  actual  tensile  strength  of  the  plates  is  not  known.  Similarly 
for  compressive  strength  they  recommend/.  =  95,000  for  mild  steel. 
Late  editions  of  Code  give  the  above  maximum  values  only. 

67.  Strength,  Proportions,  and  Efficiency  of  Joints. — No 
riveted  joint  can  be  as  strong  as  the  unperforated  plate.  The 
ratio  of  strength  of  joint  to  strength  of  unperforated  plate  is 
called  the  JOINT  EFFICIENCY. 

As  stated  in  §  65  the  highest  efficiency  for  a  joint  is  obtained 
when  the  relations  between  thickness  of  plate,  diameter  of  rivet, 
pitch,  and  margin  are  such  that  the  tendency  for  the  joint  to 
fail  in  any  one  way  does  not  exceed  the  tendency  for  it  to  fail 
in  any  other  way.  Formulae  can  be  developed  for  rinding  their 
proper  values  for  each  form  of  joint. 

Let  d  =  diameter  of  rivet-hole  in  inches;  rg-''  >  rivet; 
a  =  pitch  of  rivets  in  inches ; 
t  =  thickness  of  plates  in  inches ; 

ft  =  tensile  strength  of  plates  in  pounds  per  square  inch ; 
fc  =  crushing  strength  of  rivets  or  plates,  if  rivets  are  in 

single  shear,  pounds  per  square  inch ; 
//=  crushing  strength  of  rivets  or  plates,  if  rivets  are  in 

double  shear,  pounds  per  square  inch; 
/,  =  shearing  strength  of  rivets  in  single  shear,  pounds  per 

square  inch; 
//  =  shearing  strength  of    rivets    in    double  shear,  pounds 

per  square  inch. 

Each  joint  may  be  treated  as  if  made  up  of  a  successive  series 
of  similar  strips,  each  unit  strip  having  a  width  equal  to  a,  the 
distance  between  centers  of  two  consecutive  rivets  in  the  same 


MACHINE    DESIGN. 


row  (see  Fig.  62).  If  the  stresses  and  proportions  for  one  such 
strip  are  determined,  the  results  obtained  will,  of  course,  apply 
to  all  of  the  others,  and  consequently  to  the  whole  joint.  Con- 
sider such  a  strip  of  thickness  /  and  width  a. 


©    O 
o 


FIG.  62. 

Let  P  =  ultimate  tensile  strength  of  unperf orated  strip,  pounds; 
T=  ultimate  tensile  strength  of  net  section  of  strip,  pounds; 
5=  ultimate  shearing  resistance  of  all  rivets  in  strip, 

pounds ; 
C  =  ultimate  crushing  resistance  of  all  rivets  or  sides  of 

holes,  pounds; 
E=  efficiency  of  joint. 

To  illustrate  this  method,   consider  first  the  simplest  joint, 
i.e.,  the  single-riveted  lap-joint. 

The  unperforated  strip  has  a  tensile  strength 


(i) 


Along  the  row  of  rivets  the  net  width  of  plate  is  less  than  the 
total  width  of  the  strip  by  an  amount  equal  to  the  diameter  of 
the  rivet,  and  consequently  the  net  tensile  strength  of  the  strip 
is  expressed  by  the  equation 


(2) 


In  each  unit  strip  there  is  but  a  single  rivet  with  but  one  sur- 
face in  shear,  hence 

7T 
4- 


(3) 


RIVETED  JOINTS.  il$ 

The  crushing  resistance  of  the  rivet,  or  of  the  plate  around 
the  hole,  may  be  written  as 

C=dtfe  .........     (4) 

For  highest  efficiency  T  =  S  =  C. 
Equating  S  and  C,  (3)  and  (4),  .fjS^ 


(5) 


This  equation  gives  the  proper  theoretical  value  of  d  for  a 
given  value  of  /,  and  for  materials  represented  by  jc  and  }8. 
Equating  T  and  5,  (2)  and  (3), 


0  - 

.'.  a-^/'  +  dL     ......     (6) 

*/l 

This  gives  the  proper  theoretical  pitch.  The  efficiency  of  the 
joint  is  obtained  by  dividing  T,  S,  or  C  by  P. 

In  most  cases  the  values  of  d  and  a  as  determined  by  (5) 
and  (6)  cannot  be  strictly  adhered  to.  Stock  sizes  of  rivets 
must  be  used  in  practice,  and  there  are  also  limitations  connected 
with  the  largest  sizes  it  is  convenient  to  drive.  These  equations, 
furthermore,  do  not  take  into  consideration  the  stresses  set  up  in 
the  rivets  when  their  shrinkage,  due  to  cooling,  is  resisted  by 
the  plates,  an  item  which  may  become  excessive  with  the  smaller 
diameters.  The  spacing  of  the  rivets  must  also  be  modified  quite 
frequently  by  the  proportions  of  the  parts  to  be  connected,  by 
allowance  for  proper  space  to  form  the  heads,  and  by  provision 
for  tightness.  In  practice  it  is  therefore  often  necessary  to  depart 
from  these  values. 

It  must  be  borne  in  mind,  however,  that  any  departure  from 
the  values  of  d  and  a  given  in  (5)  and  (6)  destroys  the  equality 


Il6  MACHINE  DESIGN. 

between  T,  5,  and  C,  and  if  such  departure  is  made,  the  actual 
value  of  T,  5,  and  C  should  be  determined  (by  substitution  of 
the  values  of  d  and  a  decided  upon).  The  efficiency  of  the 
joint  will  then  be  found  by  dividing  whichever  has  the  smallest 
value,  T,  S,  or  C,  by  P. 

If  the  REAL  efficiency  of  the  joint  is  desired,  the  value  of 
T  must  be  obtained  by  increasing  }t  by  the  amount  called  for  by 
the  perforation  of  the  plates.  As  explained  in  §  63  this  will  be 

(  2  +—7=)  (  ^--  )  per  cent  greater  than  the  ft  of  the  original,  un- 
\  v  /  /  \  2-5  ' 

perforated  plate. 

68.  Problem.  —  The  following  problem  illustrates  the  method 
of  using  Table  II  in  connection  with  the  formulae  (5)  and  (6). 

What  should  be  the  dimensions  of  rivet-hole  and  pitch  for  a 
single-riveted  lap-joint  for  f-inch  iron  plates  using  iron  rivets? 

Table  II  gives  as  values  of  //,  /„  and  }c  40,000,  38,000,  and 
67,000  Ibs.  per  square  inch,  respectively,  for  this  form  of  joint. 


Substituting  these  values,  equation  (5)  becomes 


and  equation  (6) 

.  7854  X.842X  38000 

a  =  ^-^  —  +  .84  =  2.24  inches. 

.375X40000 

69.  Proportions  of  Single-riveted  Lap-joints.  —  Table  III 
and  Table  IV  have  been  computed  in  this  way.  As  Table  IV 
refers  to  steel  joints,  the  values  of  ft,  /„  and  fe  are  55,000,  47>5°°> 
and  85,000  Ibs.  per  square  inch,  respectively. 


RIVETED  JOINTS. 


117 


TABLE  III. — PROPORTIONS  OF  SINGLE-RIVETED  LAP-JOINTS,  IRON  PLATES,  AND 
RIVETS,  PUNCHED  HOLES. 


t 

!c 

.7854^5 

d 

a 

•27tfs 

tit 

% 

.42 

I  .  12 

•56 

i-49 

'\ 

k 

.70 

!.86 

f 

if 

.84 

2.24 

H 

if  -2 

I  .  12 

2.98 

f~ri 

2-2rV 

I.4O 

3-73 

H 

1.68 

4-47 

i—  1£ 

"2\-2\ 

i 

1.96 

5-22 

i-1! 

2^-3 

2.24 

5-96 

i-ij 

2i~2j 

i* 

2.52 

6.71 

jf-if 

2f-3 

Column  i  gives  the  thickness  of  plate;  columns  2  and  3  give 
the  corresponding  calculated  values  of  d  and  a  for  joint  of  maxi- 
mum efficiency;  columns  4  and  5  give  the  values  of  d  and  a 
as  compiled  by  Twiddell  in  the  Proc.  Inst.  of  M.  E.,  1881,  pp. 
293-295,  from  boiler-makers'  practice.  It  will  be  noted  that  the 
rivets  used  in  practice  (see  column  4)  are  considerably  smaller 
in  diameter  than  those  called  for  in  column  2,  and  that  this  differ- 
ence grows  more  and  more  marked  as  the  thickness  of  the  plate 
increases.  The  reason  for  this  is  that  the  difficulty  in  driving 
rivets  increases  very  rapidly  with  their  size,  i|  or  if  inches 
being  the  largest  rivet  that  can  be  driven  conveniently.  The 
equality  of  strength  to  resist  bearing  pressure  and  shear  is  there- 
fore sacrificed  to  convenience  in  manipulation.  As  the  diameter 
of  the  rivet  is  increased  the  area  to  resist  bearing  pressure  in- 
creases less  rapidly  than  the  area  to  resist  shear  (the  thickness 
of  the  plate  remaining  the  same),  the  former  varying  as  d  and  the 
latter  as  d2',  therefore  if  d  is  not  increased  as  much  as  is  neces- 
sary for  equality  of  strength,  the  excess  of  strength  will  be  to  re- 
sist bearing  pressure.  If  the  other  parts  of  the  joint  are  made 
as  strong  as  the  rivet  in  shear,  and  this  strength  is  calculated 
from  the  stress  to  be  resisted,  the  joint  will  evidently  be  correctly 
proportioned.  As  machine-riveting  comes  into  more  general 


u8 


MACHINE   DESIGN 


use  and  pneumatic  tools  are  used  in   ".hand- work,"   this   dis- 
crepancy will  tend  to  disappear. 


TABLE  IV. — PROPORTIONS  OF  SINGLE-RIVETED  LAP-JOINTS,  STEEL  PLATES,  AND 
RIVETS,  PUNCHED  HOLES. 


t 

rf-x.,,4- 

h 

.7854^/5 

•  —  Tf—  +d 

d 

a 

& 

•43 
•57 

i.  08 
i-45 

•47 
.61 

£ 

f\ 

•7i 

1.81 

.81 

2 

I 

.86 

2.17 

•94 

*& 

* 

1.14 

2.89 

1.19 

3 

Column  i  gives  the  thickness  of  the  plate;  columns  2  and  3 
give  the  values  of  d  and  a  calculated  for  joint  of  maximum  effi- 
ciency; columns  4  and  5  give  proportions  from  practice,  the 
authority  being  Moberly  (see  Stoney,  "  Strength  and  Proportions 
of  Riveted  Joints,"  p.  80).  It  will  be  noted  how  closely  the 
theory  and  practice  agree  here  for  boiler  joints.* 

70.  Single-riveted  Butt-joints.  —  To  develop  the  general 
formulae  for  the  values  of  a  and  d  for  single-riveted  butt-joints 
with  double  cover-plates  the  same  general  method  used  in  §  67 
applies. 

In  this  case  the  rivets  are  in  double  shear.     Therefore 


"T'"  '  ' (7) 


while  T  =  (a  —  d)t}t,  (2),  as  before  and 


(8) 


Equating  5  and  C,  (7)  and  (8), 


*  For  further  data,  compiled  from  American  practice,  see  sec.  73. 


RIVE  TED  JOIN  TS.  1  1  9 

//-*/.';    .".       //-*'      ' 


I  r 

and  d  =  .64f-,t  .........     (9) 

/* 

Equating  T  and  5,  (2)  and  (7), 


For  Double-riveted  Lap-joints  the  unit  strip  contains  two  rivets, 
each  in  single  shear.    The  following  equations  cover  the  case  : 

T  =  (a-d)tft, 


71.  Double-riveted  Butt-joint.  —  For  double  riveted  butt- 
joints,  double  cover-plate,  either  chain  or  staggered  riveting,  there 
are  two  rivets  in  double  shear  for  each  «^it  strip. 

T  =  (a-d)tft, 


120  MACHINE  DESIGN. 


t.        ........    (13) 


72.  General  Formulae.  —  The  following  general  equations  for 
riveted  joints  have  been  developed  by  Mr.  W.  N.  Barnard:* 

The  unit  strip  is  of  width  equal  to  the  pitch,  the  maximum 
pitch  being  taken  unless  all  rows  have  the  same  pitch. 

The  general  expression  for  the  net  tensile  strength  of  the  unit 
strip  is 

T=(a-d)l}t.   .......    (15) 

The  general  expression  for  resistance  to  shearing  of  the  rivets 
in  the  unit  strip  is 


nnd?  .  , 

5—  j-/.+-      -//,      .....    (16) 
4  4 

in  which  n  equals  the  number  of   rivets  in  single  shear  and  m 
equals  the  number  of  rivets  in  double  shear. 

The  general  expression  for  resistance  to  crushing  of  the  unit 
strip  is 

(17). 


The  tensile  resistance  of  the  solid  strip  is 

P  =  atft  ...........     (18) 

Equating  5  and  C,  (16)  and  (17),  and  transposing,  we  get 


*  See  article,  "  General  Formulas  for  Efficiency  and  Proportions  for  Riveted 
Joints,"  by  Professor  J.  H,  Barr  in  Sibley  Journal  of  Engineering,  Oct.,  1900. 


'RIVETED  JOINTS.  121 

Equating  T  and  C,  (15)  and  (17), 

(20) 


Or,  equating  T  and  S,  (15)  and  (16), 


The  following  equation   for  efficiency  has    been   developed 

on  the  assumption  that  T  =  S  =  C. 

c 
From  E-  =  —  we  get,  by  substitution  and  transposition, 


(22) 


This  equation  is  useful  in  finding  the  limiting  efficiency  oi 
joint  for  any  form  and  materials;  the  actual  proportions  adopted 
may  give  a  lower  efficiency,  but  can  never  give  a  higher  efficiency.* 
73.  Proportions  of  Joints.  —  In  American  practice  it  will  be 
found  that  there  is  more  or  less  departure  from  the  proportions 
which  would  be  arrived  at  by  the  strict  application  of  the  prin- 
ciples laid  down  in  the  preceding  articles.  This  variation  is 
due  to  several  considerations.  Chief  among  them  is  the  practical 
difficulty  of  driving  large  rivets,  thus  leading  to  the  adoption  of 
rivet  diameters  with  reference  to  convenience  of  manipulation 
rather  than  efficiency  of  joint.  As  machines  displace  handwork 
the  reason  for  this  departure  disappears  and  there  is  an  increasing 
tendency  to  use  the  larger  and  more  correct  rivet  diameters. 
Conservatism  must  be  reckoned  with  here  and  also  in  the  failure 
to  recognize  the  fact  that  rivet  diameters  do  not  depend  solely 

*  In  the  Proceedings  of  the  Inst.  of  M.  E.,  1881,  there  is  an  article  entitled 
"On  Riveting,  with  Special  Reference  to  Ship-work,"  M.  Le  Baron  Clauzel, 
which  enters  deeply  into  the  development  of  general  formulae. 


122  MACHINE  DESIGN. 

upon  the  thickness  of  plates,  but  also  should  vary  with  the  kind 
of  joint.  Practice  tends  to  hold  to  one  diameter  of  rivet  for  each 
thickness  of  plate,  irrespective  of  the  kind  of  joint. 

Another  item  of  practical  importance  is  tightness  against 
leakage  under  pressure.  Most  formulae  are  developed  without 
consideration  of  this  important  factor.  From  a  practical  point 
of  view,  the  joint  fails  when  it  begins  to  leak;  actual  rupture 
need  not  take  place.  The  topic  of  the  allowable  maximum 
pitch  as  governed  by  experience  with  tightness  of  joints  is 
discussed  in  §  80. 

The  margin  in  a  riveted  joint  is  the  distance  from  the  edge  of 
the  sheet  to  the  rivet  hole.  This  must  be  made  of  such  value  that 
there  shall  be  safety  against  failure  by  the  rivet  tearing  out. 
There  can  be  no  satisfactory  theoretical  determination  of  this 
value;  until  recently  it  has  been  held  that  practice  and  experi- 
ments with  actual  joints  showed  that  a  joint  would  not  yield  in 
this  way  if  the  margin  were  made  =  d  =  diameter  of  the  rivet 
hole.  This  is  a  safe  rule  for  iron  rivets  in  steel  plates  for  any 
type  of  joint.  Where  steel  rivets  are  used  it  will  be  well  to 

increase  this  to  -  d. 

4 
The  American  Machinist,  May  3,  1906,  says:   The  minimum 

distance  from  the  center  of  any  rivet  hole  to  a  sheared  edge 
ought  not  to  be  less  than  \\"  for  J"  rivets,  ij"  for  f "  rivets,  i J" 
for  f"  rivets,  i"  for  J"  rivets;  and  to  rolled  edges  i}",  if",  i", 
and  f",  respectively.  The  maximum  distance  from  any  edge 
should  be  eight  times  the  thickness  of  plate. 

The  distance  between  the  center  lines  of  rows  may  be  taken 
not  less  than  2.5^  for  double-chain  riveting,  and  i.88d  for 
double-staggered  riveting.  This  will  insure  safety  against  zig- 
zag tearing  of  the  plate,  but  brings  the  heads  very  close  together. 
It  is  customary  to  use  2.5 d  to  2. 75^  for  both.  From  these  values 
and  those  of  margins,  as  just  discussed,  the  proper  amount  of  lap 
can  readily  be  determined  for  any  kind  of  joint. 


RIYETED  JOINTS. 


123 


74.  Relative  Efficiencies  of  Various  Kinds  of  Joints. — The 

actual  efficiencies  of  joints  when  tested  show  some  departure 
from  the  calculated  ideal  efficiencies.  The  following  Table  (V) 
has  been  compiled  from  the  results  of  tests  to  show  roughly  the 
relative  efficiencies  of  various  types  of  joints : 

TABLE  V. — RELATIVE  EFFICIENCY  OF  IRON  JOINTS. 


Efficiency 
Per  Cent. 


Original  solid  plate 

Lap-joint,  single- riveted,  punched 

drilled 

double     "      

Butt-  oint,  single  cover,  single-riveted. . 
' '         "       double-riveted, 
double    ' '       single-riveted .  . 
"        "       double-riveted. 


IOO 

45 


45-50 
60 

55 
66 


RELATIVE  EFFICIENCY  OF  STEEL  JOINTS. 


Efficiency  Per  Cent. 


Thic 
i-f 

kness  of  PI 

M 

ates. 
i-i 

Original  solid  plate 

IOO 

IOO 

IOO 

Lap-joint   single-riveted   punched 

tJO 

Ai 

40 

'  '                    '  '               drilled          ... 

cir 

fQ 

4? 

'  '         double-riveted  punched  

7e 

7O 

65 

drilled  

80 

75 

70 

Butt-joint,  double  cover,  single-riveted,  drilled  

70 

65 

60 

"                "          "       double-riveted,  punched.  .  . 

75 

70 

65 

drilled  

80 

75 

70 

These  tables  are  from  Stoney's  "Strength  and  Proportions  of  Riveted  Joints." 

Triple  riveted  butt-joints  with  double  cover-plates  show 
efficiencies  ranging  from  80  to  90  per  cent.* 

Quadruple  joints,  of  this  form,  range  from  90  to  95  per  cent; 
and  quintuple,  from  95  to  98  per  cent. 


*  For  details  of  joints  tested,  see  Tests  of  Metals,  Watertown  Arsenal,  1896. 


1 24  MACHINE  DESIGN. 

75.  Slippage. — At  about  25  to  35  per  cent  of  its  ultimate  load 
SLIPPAGE  takes  place  in  a  riveted  joint.  This  is  probably  due 
to  the  fact  that  at  this  load  the  friction  between  the  plates,  owing 
to  the  pressure  exerted  on  them  by  the  rivets,  is  overcome.  It 
has  been  found  the  larger  the  cross-sectional  area  of  the  rivet 
the  greater  the  percentage  of  ultimate  load  which  can  be  with- 
stood without  slippage.  It  has  also  been  found  that  large  rivet- 
heads  are  better  than  small  ones  for  the  same  reason. 

The  importance  of  the  consideration  of  slippage  has  been  fully 
established  by  the  work  of  Professor  Bach  ("  Die  Maschinen- 
elemente,"  gth  ed.  pp.  164-195).  His  careful  and  exhaustive 
experiments  prove  that : 

1.  In  cooling   the  rivet  shrinks  away  from  the  walls  of  the 
hole. 

2.  In  consequence  of  this,  there  is  no  tendency  to  shear  off 
the  rivet  until  after  the  joint  has  failed,  for  all  practical  purposes, 
by  losing  tightness  because  of  slippage. 

3.  The  percentage  of  the  ultimate  or  rupture  load  at  which 
slippage  takes  place  varies  according  to  three  items  : 

a.  It  is  directly  proportional  to  the  square  of  the  diameter 

of  the  rivet.     From  this  the  desirability  of  using  large 
rivets  rather  than  small  is  further  established. 

b.  It  is  increased  by  calking,  especially  if  both  rivet  heads 

are  calked  as  well  as  the  plate  edges. 

c.  It  is  greatly  increased  by  holding  the  rivets  under  maxi- 

mum pressure  until  they  are  cool  enough  to  have  set. 

This  gives   better   results   than   blows,  light   pressure, 

or  early  removal  of  pressure. 

Professor  Bach  argues  that  joints  should  not  be  proportioned 
with  reference  to  the  ultimate  or  rupture  strength.  He  claims 
that  the  maximum  pitch  is  determined  by  the  condition  of  tight- 
ness against  springing  open  between  rivets  when  pressure  is 
applied.  The  minimum  pitch  is  that  fixed  by  the  spacing 
of  rivet  heads  which  is  the  least  which  will  permit  calking  them. 
Between  these  limits  he  chooses  pitch: 


For  Single.  Double,  Treble  Riveted  Lap  "Joint. 
For  Single  RLveteld  Double  Butt  Strap  Jojnt 
.   '•    Double'  «  «          "        ."'       (" 

»*    Treble     «'  «*          «        "  -      «' 


I**- 


IX' 


w 


w 


X- 


X" 


IX 


Thickness  of  Plates-T. 
Fig.  1.    RIVET  DIAMETERS  BY  BACH'S  FORMULA, 


FJg.  3.    SINGLE  RIVETED  LAP  JOINT. 


Fig.  4.    DOUBLE  RIVETED  LAP  JOINT. 


Fifl-7.    DOUBLE  RIVETED  DOUBLE  B.UTT  STRAP  JOINT.  Fifi..8,    DOUBLE  RIVETED  DOUI 


PLATE  L 


s 

o  5* 

a 


: 


5"  6' 

Fig.  2.    RIVET  LENGTH  FOR  GIVEN  PLATE  THICKNESS 


<>''  37/  4'' 

Thickness  of  Plates-T. 


Fig.  5.    DOUBLE  RIVETED  LAP  JOINT. 


Fig.  6.    TREBLE  RIVETED  LAP  JOINT. 


BUTT  STRAP  JOINT.-*  F\g.  9.    DOUBLE  RIVETED  QOUBLE  BUTT  STRAP 


R1YETED  JOINTS. 


125 


1.  So  that  the  safe  resistance  to  slippage  (as  experimentally 
determined  by  him)  is  equated  to  the  stress  on  the  joint  due  to 
the  diameter  of  the  vessel  and  the  pressure. 

2.  So  that  the  unit  stress  in  the  plate  between  the  rivets  shall 
not  exceed  the  safe  working  value  of  the  plate  material  when  the 
strength  of  the  perforated  section  is  equated  to  the  stress  due  to 
the  diameter  and  pressure. 

Plate  I  shows  graphically  the  proportions  of  riveted  joints  as 
determined  from  Professor  Bach's  formulas  by  M.  Shibata  in 
the  American  Machinist,  Vols.  26  and  27. 

76.  Rivet  Size  and  Proportions. — In  general  the  rivet  should 
have  a  shank  A  inch  smaller  in  diameter  than  the  hole  to  be 
filled,  while  the  head  should  have  a  diameter  of  from  1.6  to  2 
times  the  diameter  of  hole,  and  a  height  of  from  .6  to  .75  times 
the  hole  diameter.  Especial  care  should  be  taken  in  the  case 
of  machine-riveting  to  have  just  enough  metal  projecting  beyond 
the  hole  to  allow  for  the  necessary  upset  for  the  shank  to  fill 
the  hole,  with  just  enough  left  over  to  fill  the  die  for  the  head. 


-E— 

ABC 
FIG.  63. 

TABLE  OF  DIMENSIONS  OF  RIVET  HEADS. 


Diameter 

Pan  Head. 

Button  Head. 

Counter  Sunk. 

of  Rivet. 

A 

B 

C 

d 

E 

F 

G 

E 

G 

E 

G 

ft 

* 

if 

jl 

r* 

4 

t 

iA 
i* 

g 

f 

«i 

H 

i± 

& 

if 

| 

i 

•ft 

t 

A 

f 

if 

ik 

I 

if 

H 

if 

1 

if 

* 

126  MACHINE  DESIGN. 

77.  Problem. — How  far  must  the  tail  of  the  rivet  project  in 
order  to  satisfy  the  above  conditions  for  the  following  case:  Two 
plates  f  inch  thick  each  are  to  be  connected,  using  J-inch  rivets 
in  H-inch  holes.     The  head  is  to  be  cone-shaped,  having  an  out- 
side diameter  of  if  inches  and  a  height  of  J  inch. 

The  cubical  contents  of  the  cone  head  =  area  of  baseXj 
altitude  =  2. 76  square  inches X. 25  inch  =  .69  cubic  inch. 

The  difference  in  cubical  contents  between  a  hole  it  inch  in 
diameter  by  j  inch  long  and  a  shank  J  inch  in  diameter  and 
f  inch  long  =  fX-7854(i|2—J2)=. 067  cubic  inch. 

The  amount  required  for  head  and  upset  therefore  equals 
.69 +  .067  =  .757  cubic  inch. 

The  area  of  the  f -inch  shank  =  .60  square  inch.  .757  cubic  inch, 

therefore,  calls  for  a  length  of  --^—=1.25  inches.     This  amount 

,00 

would  then  be  the  projection  through  the  plate.     The  length  of 
rivet-shank  called  for  would  equal  f  inch+i}  inches  =  2  inches. 

NOTE. — Had  the  head  been  cup-shaped,  its  cubical  contents 
should  have  been  taken  as  that  of  a  spherical  segment.  For  cup- 
shaped  heads  the  diameter  is  about  i.  7  X  diameter  of  hole,  and 
the  height  about  .6  X  diameter  of  hole.  The  volume  of  the 
spherical  segment  is  given  by  the  following  rule:  Multiply  half 
the  height  of  the  segment  by  the  area  of  the  base  and  the  cube 
of  the  height  by  .5236  and  add  the  two  products. 

78.  Countersunk  Rivets. — Fig.  63  C  shows  the  proportions  for 
a  countersunk  rivet.     C6untersunk  rivets  make  a  much  weaker 
and  less  reliable  joint  than  the  ordinary  form,  and  should  only 
be  used  where  it  is  absolutely  necessary  that  the  surface  of  the 
plate  be  free  from  projections. 

79.  Nickel-steel  Rivets. — Where  peculiar  conditions   call  for 
great  strength  of  rivet  combined  with  small  area,  it  may  be  found 
desirable  to  use  nickel-steel  rivets.     Experiments  made  by  Mr. 
Maunsel  White  (see  Journal  Am.  Soc.  of  Nav.  Eng.,  1898)  on 


RIVETED  JOINTS.  127 

riveted  joints  using  nickel-steel  rivets  showed  an  average  shear- 
ing resistance  of  85,720  Ibs.  per  square  inch  for  single  shear,  and 
an  average  of  90,075  Ibs.  per  square  inch  for  double  shear.  These 
values,  it  will  be  noted,  are  nearly  double  those  of  the  very  mild 
steel  ordinarily  used.  The  rivets  were  f  inch  in  diameter,  and 
some  of  the  joints  failed  by  tearing  the  plates,  while  others  failed 
by  shearing  the  rivets.* 

80.  Construction  of  Tight  Joints.  —  In  general   three  types 
of  riveted  joints  may  be  recognized : 

1.  Those  in  which  strength  is  the  sole  factor  of  importance, 
as  in  most  purely  structural  iron  and  steel  work. 

2.  Those  in  which  strength  and  tightness  are  equally  deter- 
mining elements,  as  in  boilers  and  pressure  pipes. 

3.  Those  in  which  tightness  is  the  prime  consideration,  as 
in  tanks  subjected  to  only  light  pressure. 

In  punching  holes  in  plates,  it  is,  of  course,  necessary  to 
have  the  hole  in  the  die-block  larger  than  the  punch.  The 
consequence  is  that  the  holes  are  considerably  tapered  and 
care  should  be  exercised  in  joining  the  plates  that  the 
small  ends  of  the  holes  be  together  as  shown  in  Fig.  64^, 
and  not  apart  as  shown  in  Fig  64$.  It  is  obvious  that  at 
A  the  pressure  on  the  rivet  tends  to  draw  the  plates  closer 
together,  and  that  as  the  rivet  cools  its  longitudinal  shrinkage 
will  tend  to  keep  it  a  tight  fit  for  the  hole  in  spite  of  its  diametral 
shrinkage. 

It  is  equally  obvious  that  at  B  the  pressure  on  the  rivet  will 


*  See  Bulletin  No.  49,  Eng.  Exp.  Station,  University  of  Illinois,  for  exhaustive 
tests,  by  Professors  Talbot  and  Moore,  on  nickel-steel  riveted  joints,  undertaken  at 
the  request  of  the  Board  of  Engineers  of  the  Quebec  Bridge  and  the  Pennsylvania 
Steel  Co.  The  chief  conclusions  to  be  drawn  are  that,  while  these  joints  show 
advantage  over  ordinary  carbon  steel  as  concerns  ultimate  strength,  there  is  no 
such  advantage  as  regards  slippage.  All  of  the  joints  tested  failed  by  shearing  the 
rivets.  This  shows  that  the  rivets  were  theoretically  too  small  and  accounts  for 
the  slippage.  See  Sec.  75. 


128  MACHINE    DESIGN. 

tend  to  force  the  plates  apart  and  squeeze  metal  between  them, 
and  also  that  all  shrinkage  of  the  rivet  will  be  away  from  the 
walls  of  the  hole. 

In  using  drilled  plates  care  must  be  exercised  to  remove  the 
sharp  burrs  left  by  the  drill,  as  experience  has  shown  that  this 
has  a  considerable  effect  on  the  strength  of  the  joint. 

Where  the  plates  form  the  walls  of  vessels  to  hold  fluids,  the 
joints  must  be  designed  with  a  view  toward  tightness  as  well  as 


FIG.  64.  FIG.  65. 

strength.  For  this  purpose  the  edges  are  planed  at  a  slight  bevel, 
and  calked  as  shown  in  Fig.  65  by  a  tool  which  resembles  a  cold- 
chisel  with  a  round  nose.  Pneumatic  tools  are  used  for  this  pur- 
pose almost  entirely,  as  they  execute  more  uniform  and  rapid 
work  than  can  be  done  by  hand.  In  calking  great  care  should 
be  exercised  not  to  groove  the  plates  at  A- A,  as  these  are  danger- 
points  for  bending,  and  an  incipient  groove  is  very  apt  to  develop 
into  a  crack.  It  is  largely  on  this  account  that  the  round- 
lose  calking-tool  has  superseded  the  square-nose  in  the  best 
practice. 

It  has  been  found  that  the  load  which  the  joint  will  carry 
before  leaking  is  greatly  increased  if  both  rivet  heads  are  calked 
as  well  as  the  plate  edges.  .  (Bach's  experiments.) 

The  consideration  of  tightness  has  a  determining  effect  on  the 
maximum  allowable  pitch  for  any  given  thickness  of  plate  and 
type  of  joint.  Based  upon  practice  the  following  values  have 
been  found  safe  for  r5^  "  plates: — 

Single  riveted  lap  joints,  pitch  =  ;t 

Double  riveted  lap  joints,  pitch  =  p^t 

Double  riveted  butt  joints,  pitch  (in  outer  row)  =  14.5! 

Triple  riveted  butt  joints,  pitch  (in  outer  row)   =  2ot. 


KIVETED  JOINTS.  129 

Because  of  the  use,  with  heavier  plates,  of  rivet  diameters 
which  are  proportionately  too  small,  these  ratios  of  pitch  to 
thickness  of  plate  will  be  found  to  decrease  in  practice  as  the 
thickness  of  plate  increases.  Thus  for  £"  plates  they  become 
5t,  6.6t,  10.251,  and  i6t,  respectively.* 

81.  Materials    to   be    Used. — The    material    to    be  used  in 
riveted  joints  depends,   of  course,   on  the  nature  of  the  work, 
but  in  general  it  may  be  said  that  extremely  mild  and  highly 
ductile  steel  as  free  from  phosphorus  and  sulphur  as  possible 
should  be  used.     Open-hearth  steel  is  greatly  to  be  preferred  to 
Bessemer,  f 

82.  Plates   with    Upset    Edges. — Some    boiler-makers     have 
adopted,  as  an  expedient  for  saving  material,  a  method  of  using 
plates  with  thickened  (upset)  edges.     If  we  let  t  represent  the 
thickness  of  the  body  of  the  plate  and  /'  the  thickness  of  the  edge, 
while  a  represents  the  pitch  and  d  the  diameter  of  hole,  then,  when 

•-£* 

the  joint  will  be  as  strong  as  any  other  section  of  the  plate,  the 
joint  being  proportioned,  of  course,  for  the  thickness  t' .  It  is 
customary  to  thicken  only  the  edges  which  form  the  longitudinal 
seam.  This  method  is  open  to  two  serious  objections.  Unless 
the  plates  are  very  carefully  annealed  after  being  upset  they  are 
almost  certain  to  be  weakened  by  indeterminate  working  and 
cooling  stresses.  Moreover,  although  the  original  new  joint  may 
show  as  high  an  efficiency  as  if  the  plates  throughout  were  of  the 
thickness  t',  as  corrosion  proceeds,  it  acts  more  on  the  plate  away 
from  the  joint  than  at  the  joint,  because  at  the  latter  place  the 
plate  is  protected  by  the  cover-plate  or  rivet-head  or  both.  The 

*  For  quadruple  and  quintuple  butt  joints  they  may  be  3  2t  and  48^,  respect- 
ively. 

t  Standard  specifications  can  be  found  in  the  A.S.M.E.  Boiler  Code. 


130 


MACHINE  DESIGN. 


result  is  a  shorter  life  under  full  pressure  for  the  boiler  with  thin 
plates  and  thickened  edges. 

83.  Joints  for  More  than  Two  Plates. — The  joints  considered 
thus  far  have  dealt  with  the  problem  of  connecting  the  edges 
of  two  plates  only.  In  tanks  and  boilers  which  must  have  tight 
seams  we  are  frequently  confronted  with  the  problem  of  joining 
three  and  even  four  plates.  An  instance  is  where  the  cross-seam 
and  longitudinal  seam  of  a  boiler  meet.  The  joint  is  made  by 
thinning  down  one  or  more  of  the  plates.  Figs.  66  to  70  (taken 
from  Unwin's  "Machine  Design")  show  the  methods  employed. 
Fig.  66  shows  a  junction  of  three  plates,  a,  6,  and  c,  where  both 
seams  are  single-riveted  lap-joints.  It  will  be  seen  that  the 
corner  of  a  is  simply  drawn  down  to  an  edge  and  " tucked 
under"  c. 

Fig.  67  shows  a  junction  of  three  plates  where  one  seam  is  a 
single-riveted  and  the  other  a  double-riveted  Jap-joint.  As 
before,  the  corner  of  a  is  drawn  down  and  tucked  under  c. 

Fig.  68  shows  the  junction  of  three  plates  where  both  seam 3 
are  single -riveted,  single  cover- plate  butt-joints.  The  plates 
merely  abut  against  each  other,  but  the  longitudinal  cover  is 


b               j-Q 
L               /!  ° 

i 

0    0    0    10 

\ 

ojo 

O    O    O  i  O 

11  0  0  0  0  0 

0|0 

r  o  o  o  o  o 

mo 

i-—  [s. 

iw 

0]0 

trx 

Jj 

L_L__ 

obi 

_/ 

FIG.  66. 


FIG.  67. 


FIG.  68. 


drawn  down  and  tucked  under  the  cross-seam  cover  which  is 
thinned  down  to  match. 

Fig.  69  also  shows  the  junction  of  three  plates ;  here  the  cross- 
seam  is  a  single-riveted  lap-joint  while  the  longitudinal  joint 
is  a  double-riveted  butt-joint  with  double  cover-plates.  The 


RIVETED  JOINTS.  131 

upper  cover-plate   is  planed  on  the  end  so  that  it  can  be  tightly 
calked  where  it  abuts  against  the  plate  c. 


b 

on 

s*\\ 

(f 

( 

V 

) 

f6~0^ 
3  O  O 

8 

Qic 

( 

(t 

) 

' 

POO 

[  O  O  Q 

0 
O 

a 

° 

FIG.  69. 


FIG,  70. 


A  method  of  joining  four  plates  is  shown  in  Fig.  70.  Both 
seams  are  single-riveted  lap-joints,  b  and  c  are  both  drawn 
down  as  shown. 

84.  Junction  of  Plates  Not  in  Same  Plane. — Where  the  plates 
to  be  joined  are  in  different  planes,  it  is  customary  to  use  some 
one  of  the  rolled  structural  forms.  Fig.  71  shows  the  method 
of  using  an  angle  iron  for  plates  at  a  right  angle  to  each  other. 

Where  it  is  possible  to  turn  a  flange  on  one  of  the  plates  this 
method  is  often  adopted.  Care  should  be  taken  not  to  use  too 


oooo 


FIG.  71. 


FIG.  72. 


sharp  a  radius  of  curvature  (the  inside  radius  must  be  greater 
than  the  thickness  of  the  plate  even  with  the  mildest  steel)  and 
the  flanged  plate  should  be  thoroughly  annealed  after  it  is  bent. 

Fig.  72  shows  the  method  of  making  flanged  joints  such  as 
are  frequently  used  in  connecting  boiler  heads  and  shells. 

85.  Problem. — The  following  problem  will  serve  to  illustrate 
the  design  of  riveted  joints  for  boilers.  It  is  required  to  design  a 


132  MACHINE  DESIGN. 

horizontal  tubular  boiler  48  inches  in  diameter  to  carry  a  work- 
ing pressure  of  100  pounds  per  square  inch. 

A  boiler  of  this  type  consists  of  a  cylindrical  shell  of  wrought 
iron  or  steel  plates  made  up  in  length  of  two  or  more  courses  or 
sections.  Each  course  is  made  by  rolling  a  flat  sheet  into  a 
hollow  cylinder  and  joining  its  edges  by  means  of  a  riveted  joint, 
called  the  longitudinal  joint  or  seam.  The  courses  are  joined  to 
each  other  also  by  riveted  joints,  called  circular  joints  or  cross- 
seams.  Circular  heads  of  the  same  material  have  a  flange  turned 
all  around  their  circumference,  by  means  of  which  they  are 
riveted  to  the  shell.  The  proper  thickness  of  plate  may  be 
determined  from  (I)  The  diameter  of  shell  =  48  inches;  (II)  The 
working  steam-pressure  per  square  inch  =  ioo  pounds;  (III)  The 
tensile  strength  of  the  material  used;  let  steel  plates  be  used 
of  60,000  pounds  specified  tensile  strength. 

Preliminary  investigations  of  the  conditions  of  stress  in  the 
cross-section  of  material  cut  by  a  plane  (I)  Through  the  axis; 
(II)  At  right  angles  to  the  axis,  of  a  thin  hollow  cylinder,  the 
stress  being  due  to  the  excess  of  internal  pressure  per  square  inch. 
Let  /  =the  length  of  the  cylindrical  shell  in  inches; 

J9  =  the  diameter  of  the  cylindrical  shell  in  inches; 

p  =  thc  excess  of  internal  over  external  pressure  in  pounds 
per  square  inch; 

/i  =  unit  tensile  stress  in  a    longitudinal  section  of  material 
of  the  shell  due  to  p; 

J2  =  unit  tensile  stress  in  a  circular  section  of  material  of  the 

shell  due  to  p' 
t  =  thickness  of  plate; 

ft  =  ultimate  tensile  strength  of  plate. 
All  stresses  are  in  pounds  per  square  inch. 

In  a  longitudinal  section  the  total  stress  is   equal   to  IDp, 

Dp 

and  the  area  of  metal  sustaining  it  =  2lt.     Then  /i=~7  • 


RIVETED  JOINTS.  133 

In  a  circular  section  the  total  stress  =  --  •,    and  the  area 

4 


sustaining  it  =  7r£>/,  nearly.     Then 

*     op 


Therefore  the  stress  in  the  first  case  is  twice  as  great  as  in 
the  second;  and  a  thin  hollow  cylinder  is  twice  as  strong  to 
resist  rupture  on  a  circular  section  as  on  a  longitudinal  one. 
The  latter  only,  therefore,  need  be  considered  in  determining  the 
thickness  of  plate.  Equating  the  stress  due  to  p  in  a  longitudinal 
section,  and  the  strength  of  the  cross-section  of  plate  that  sustains 

it,   we  have  lDp  =  2lt}t.   Therefore  t=f>  the  thickness  of  plate 

that  would  just  yield  to  the  unit  pressure  p.  To  get  safe  thickness, 
a  factor  of  safety  K  must  be  used.  It  is  usually  equal  in  boiler- 
shells  to  5  or  6.  Its  value  is  small  because  the  material  is  highly 
resilient  and  the  changes  of  pressure  are  gradual,  i.e.,  there  are 
no  shocks.  This  takes  no  account  of  the  riveted  joint,  which 
is  the  weakest  longitudinal  section,  E  times  as  strong  as  the 
solid  plate,  E  being  the  joint  efficiency  =0.75  if  the  joint  be 


double-riveted.       The    formula    then    becomes    t=—~r.    Sub- 

z/t-h 

stituting  values, 

6X48X100  '  ' 

/  =  —  -/  -  —  -  =0.32  inch,  say  A  inch. 
2X60000X0.75 

The  circular  joints  will  be  single  -rive  ted  and  joint  efficiency 
will  =0.50.  But  the  stress  is  only  one  half  as  great  as  in  the 
longitudinal  joint,  and  therefore  it  is  stronger  in  the  proportion 
0.50X2  to  0.75,  or  i  to  0.75.  From  this  it  is  seen  that  a  circular 
joint  whose  efficiency  is  0.50  is  as  strong  as  the  solid  plate  in  a 
longitudinal  section.  From  the  value  of  /  the  joints  may  now  be 
designed. 


134  MACHINE   DESIGN. 

Consider  first  the  cross-seam.      This  is  a  single-riveted  lap- 
joint.     Assume  drilled  holes. 
Equations  (5), 

d-l.vkt, 

Is 

and  (6),  '  -*&&+4,         ' 

apply,  while  from  Table  II  we  get  as  values  of  ft,  /«,,  fc,  60,000, 
45,000,  and  85,000  pounds  per  square  inch  respectively  for  steel 
plates  and  rivets. 

8^000 

/.  d  =  i.2jX— X. 3125  =.75 inch 

1     45000      °     J      /0 


.78^4X.7[>   X^-, 

and         a=      ~  12^x60000 —  +'75  =  I-Sl  inches,  say  iif  inches. 


The  margin  =  d  =  .  7  5  inch. 

The  lap  of  the  cross-seam  =3^  =  2.25  inches. 

The  longitudinal  seam  will  be  a  double  staggered  lap-joint. 

Equations  (n)  and   (12)  apply: 


d-ift,    and     a--'  +  d. 

Is  l]t 

From  Table  II,  ft  =60,000,  /.  =46,000,  and  }c  =  85,000; 

8^000 

say  .75, 


-.75  X46ooo  .     , 

and       a  =      >3I25X6oooo     +  -75  =  2.93  inches,  say  2$ inches. 

The  distance  between  the  rows  =  1.88^  =  1.41  inches,  say  i& 
inches.  The  total  lap  in  the  longitudinal  joint  =4.88^=3.66 
inches,  say  3^  inches. 


RIYETED  JOINTS.  135 

The  joints  are  therefore  completely  determined,  and  a  detail 
of  each,  giving  dimensions,  may  be  drawn  for  the  use  of  the  work- 
men who  make  the  templets  and  lay  out  the  sheets. 

Having  determined  the  proportion  of  the  joints,  let  these 
dimensions  be  used  to  calculate  the  actual  efficiency  of  the  longi- 
tudinal seam. 

Assume  that  the  natural  tensile  strength  of  the  unperforated 
plate  is  60,000  Ibs.  per  square  inch. 

The  excess  of  strength  of  drilled  steel  plates  in  net  section 
over  unperforated  section  (see  §  63)  is 

/       6.i24\/4-5-r\ 
2  -f- — -j-     *"* —   per  cent. 

\     v*  A  2.5  y1 

d  2.0375 

Here        I— .312510.     and     r  =  -r  =—  !— ^  =3.92; 

a        -75 

.*.  excess  due  to  perforation  =3%,  nearly. 
60,000  X  i.  03  =6 1, 800  =}t. 
j,  =46,000  from  Table  II. 
/c=85,ooo    "          "      " 

T  =  (a-d)tft  =  (2.9375-. 75)(.3i25X6i,8oo)  =42,250  Ibs. 
5  =  1.57^/a  =  1.57  X . 7 52  X 46,000 =40,625  Ibs. 
C  =  2dtlc  =  2 X .75 X .3125 X85,ooo  =  39,845  Ibs. 
P=a}tt  =  2.9375 X6o,oooX. 3125  =55,075  Ibs. 
Of  J1,  5,  and  C,  the  latter  has  the  smallest  value;  the  actual 

efficiency  of  the  joint  may  be  taken  as  =-7;=^ =  .7231;   or 

J  -  y  P      55075       '  ^ 

72.35%. 

Since  T,  S,  and  C  are  unequal  it  is  evident  that  there  has  been 
departure  from  the  conditions  for  maximum  efficiency.  There  are 
two  ways  of  restoring  this  equality,  or  at  least  diminishing  the 
inequality.  If  a  be  slightly  decreased,  T"and  P  will  be  proper- 


136  MACHINE  DESIGN. 

tionately  decreased  and  5  and  C  will  have  the  same  values  as 
before. 

Leaving  a  as  before  and  increasing  d,  increases  S  as  the 
square  of  d  and  C  as  d,  while  T  and  P  remain  as  before. 

Inspection  shows  that  T  exceeds  C  by  5.7  per  cent.  There- 
fore a  may  be  decreased  by  this  percentage,  or  .17  inch.  This 
is  approximately  A  inch  and  reduces  the  pitch  from  21!  to 
2f  inches. 

Using  this  value  of  a  gives  as  the  excess  strength  due  to  per- 
foration 4.33  per  cent. 

.*.  ft  =  62,600  Ibs., 

r  =  39,i25  Ibs., 
5  =  40,625  Ibs., 
C  =  39,845  Ibs., 
^  =  51,560  Ibs., 


The  second  method  of  balancing  T,  5,  and  C  would  be  by 
increasing  d.  The  next  commercial  size  above  j  inch  would 
be  H  inch.  Leave  a  =  2$  inches,  and  increase  d  to  ft  inch,  and 
first  calculate  the  excess  strength  due  to  perforation. 


Using  these  values,  the  excess  =  4.  5  7  per  cent. 

/«  =  62,750  Ibs., 
T  =  4ij6'jo  Ibs., 
5  =  47,675  Ibs., 
C  =  43,i65  Ibs., 

p  =  5-5>°75  lbs-» 
and,  since  T  is  least, 


RIVETED  JOINTS.  137 

The  best  result  is  that  obtained  by  keeping  J  =  f  inch,  but 
changing  a  to  2f  inches,  and  it  would  be  advantageous  to  make 
these  the  proportions  of  the  joints  rather  than  those  first  de- 
termined. 

The  next  step  is  to  check  back,  using  the  proportions  de- 
cided upon,  for  the  actual  factor  of  safety  which  should  not  be 
less  than  5. 

E7"  7~)    A 

From  the  equation  (p.  133)  t  =  ——>•>  we  have 

K_zJtEt 

2X626ooX.7588X.3i25 
48X100 

Attention  should  be  called  to  the  fact  that  the  ordinary  and 
less  correct  method  of  calculating  efficiencies  ignores  the  excess 
strength  due  to  perforation,  and  the  efficiency  is  simply  taken 

T_ 

With  #  =  2lf  inches,  and  d  =  %  inch,  this  would  give  us 

=  74-47%>  instead  of  72.35%. 

With  a  =  2.\  inches  and  d  =  \  inf:h,  it  would  give  us 
72.71%   instead  of  75.88%. 


With  0=2}f  inches  and  d=¥k  inch,  it  would  give 
=  7I'97%  lnStead  °f  75*66%' 


138  MACHINE  DESIGN. 

In  practice  the  designer  must  be  familiar  with  the  code  of 
rules,  governing  all  details  of  boiler  design,  which  prevails  where 
the  boiler  is  to  be  used.  There  are  many  such  codes — private, 
state,  and  national,  and  they  are  not  in  agreement. 

One,  recently  prepared  by  a  committee  of  the  American  Society 
of  Mechanical  Engineers,  will,  it  is  to  be  hoped,  supersede 
them  all  for  the  entire  United  States. 


CHAPTER  VIII. 


BOLTS     AND     SCREWS. 

86.  Classification  and  Definition. — Bolts  and  screws  may  be 
classified  as  follows:  I.  Bolts;  II.  Studs;  III.  Cap-screws,  or 
Tap-bolts;  IV.  Set-screws;  V.  Machine  screws;  VI.  Screws 
for  power  transmission. 

A  "bolt"  consists  of  a  head  and  round  body  on  which  a 
thread  is  cut,  and  upon  which  a  nut  is  screwed.  When  a  bolt 
is  used  to  connect  machine  parts,  a  hole  the  size  of  the  body  of 
the  bolt  is  drilled  entirely  through  both  parts,  the  bolt  is  put 
through,  and  the  nut  screwed  down  upon  the  washer.  (See 
Fig-  73-) 


FIG.  73. 


FIG.  74. 


FIG.  75. 


A  "stud"  is  a  piece  of  round  metal  with  a  thread  cut  upon 
each  end.  One  end  is  screwed  into  a  tapped  hole  in  some  part 
of  a  machine,  and  the  piece  to  be  held  against  it,  having  a  hole 
the  size  of  the  body  of  the  stud,  is  put  on  and  a  nut  is  screwed 
upon  the  other  end  of  the  stud  against  the  piece  to  be  held.  (See 

Fig.  74-) 

139 


140  MACHINE  DESIGN.  "   i 

A  "cap-screw"  is  a  substitute  for  a  stud,  and  consists  of  a 
head  and  body  on  which  a  thread  is  cut.  (See  Fig.  75.)  The 
screw  is  passed  through  the  removable  part  and  screwed  into  a 
tapped  hole  in  the  part  to  which  it  is  attached.  A  cap-screw 
is  a  stud  with  a  head  substituted  for  the  nut. 

A  hole  should  never  be  tapped  into  a  cast-iron  machine 
part  when  it  can  be  avoided.  Cast  iron  is  not  good  material 
for  the  thread  of  a  nut,  since  it  is  weak  and  brittle  and  tends  to 
crumble.  In  very  many  cases,  however,  it  is  absolutely  neces- 
sary to  tap  into  cast  iron.  It  is  then  better  to  use  studs  if  the 
attached  part  needs  to  be  removed  often,  because  studs  are  put 
in  once  for  all,  and  the  cast-iron  thread  would  be  worn  out 
eventually  if  cap-screws  were  used. 

The  form  of  the  United  States  standard  screw-thread  is 
shown  in  Fig.  76.  The  sides  of  the  thread  make  an  angle  of 

60°.  Instead  of  coming  to  a  sharp 
point,  the  threads  have  a  flat  at  top 
and  bottom  whose  width  is  =  J/?,  p 
being  the  pitch.  Table  VI  gives  the 
FIG.  76.  standard  proportions. 

For  single  threads  the  lead  of  the  thread  helix  equals  />,for  double 
and  triple  threads  it  equals  2p  and  3^,  respectively.  If  clockwise 
rotation  of  the  screw  causes  the  thread  to  enter  the  nut,  the  thread 
is  termed  right-hand;  if  counter-clockwise,  left-hand. 

When  one  machine  part  surrounds  another,  as  a  pulley -hub 
surrounds  a  shaft,  relative  motion  of  the  two  is  often  prevented 
by  means  of  a  "  set-screw,"  which  is  a  threaded  body,  pref- 
erably non-projecting  (Fig.  77).  The  end  is  either  rounded  as 
in  Fig.  77  a,  or  pointed  as  in  Fig.  77  b,  or  cupped  as  in  Fig. 
77  c,  and  is  forced  against  the  inner  part  by  screwing  through 
a  tapped  hole  in  the  outer  part. 

Data  relative  to  the  holding  power  of  set-screws  will  be 
found  in  §  109. 


BOLTS  AND  SCREWS. 


141 


TABLE  VI.'— U.  S.  STANDARD  SCREW-THREADS. 


Bolts  and  Threads. 


Hex.  Nuts  and  Heads. 


Nan 


ndH. 


a 


Di 


I 

** 

trS 


jh 


S  . 


Ins. 
ft 

1 

ft 


If 
If 

a 


2f 

3 
3* 

S 

4i 
4j 
4f 

5 


5f 
6 


20 

18 
16 
14 
13 

12 
II 
10 

9 

8 

7 

6 
6 
S* 


4* 
4' 

I 

3i 
3 


Ins. 
•185 
.240 
.294 

•  344 
.400 

•454 
•507 
.620 

•731 
-837 

.940 
.065 
.160 
.284 
•389 

-491 
.616 
.712 
.962 

2.  176 

2.426 
2.629 
2.879 
3-100 
3-3I7 


3.798 
4.028 
4.256 
4.480 

4-730 

4-953 
5-203 


Ins. 

.0062 

.0074 

.0078 

.0089 

.0096 

.0104 
.0113 
.0125 
.0138 
.0156 

.0178 
.0178 
.0208 
.0208 
.0227 

.0250 
.0250 
.0277 
.0277 
.0312 

.0312 

•0357 
•0357 
.0384 
.0413 

.0413 

•0435 
•0454 
.0476 
.0500 

.0500 
.0526 
.0526 
•0555 


Sq.  Ins.  Sq.  Ins. 


Ins. 


.049 
.077 
.  no 
.150 
.  196 

.249 

•307 
.442 
.601 

.785 

•994 
1.227 

1-485 
1.767 
2.074 

2.405 
2.761 


4.909 

5-940 
7.069 
8.296 
9.621 
11.045 

12.566 
14.186 
15.904 
17.721 

I9-635 

21.648 

23-758 
25.967 
28.274 


.027 

•045 
.068 

•093 
.126 

.162 

.202 
.302 
.420 
•550 

•694 

•893 

1-057 

1-295 


1.746 
2.051 
2.3O2 

3-7I9 

4.620 
5-428 
6.510 
7-548 
8.641 

9-963 
H.329 
12-753 
14.226 

I5-763 

I7-572 
19.267 
21.262 

23.098 


Ins. 

S 


Ins. 
ft 


9ft    JC 


:! 


*i 


O4 

3l 

4 
4i 

;f 

5 

Si 
5| 

6 


i 


142 


MACHINE   DESIGN. 


The  term   "machine  screws"  covers  many  forms  of  small 

screws,  usually  with  screw-driver  heads.  All  of  the  kinds  given 

in  this  classification  are  made  in  great  variety  of  size,   form, 
length,  etc. 


FIG.  77.  FIG.  78. 

Thus  far  American  manufacturers  have  failed  to  agree  upor> 
standard  dimensions  for  set-screws  and  machine  screws.* 

For  consideration  of  their  design,  we  will  divide  bolts  and 
screws  into  three  classes : 

(a)  Those  which  are  put  under  no  stress  by  screwing  up. 

(b)  Those  which  are  put  under  an  initial  stress  by  tightening. 

(c)  Those  which  are  used  to  transmit  power. 

87.  Analysis  of  Action  of  Screw. — Before  taking  these  cases 
up  in  detail  it  will  be  well  to  examine  into  the  general  action 
of  screw  and  nut.  Reference  is  made  to  Fig.  79.  The  turning 
of  a  nut  loaded  with  W  Ibs.  may  be  considered  as  equivalent 
to  moving  a  load  W  on  an  inclined  plane  whose  angle  with 
the  horizontal  is  the  same  as  the  mean  pitch  angle  of  the 
thread  a. 

*  The  report  of  the  committee  of  the  A.  S.  M.  E.  on  this  subject  with  suggested 
standards,  will  be  found  in  Vol.  28  of  the  Trans.  A.  S.  M.  E. 


BOLTS  AND  SCREWS. 


143 


Let  r\=  outside  radius  of  thread; 
r2  =  inside  radius  of  thread; 

r  =  mean  radius  of  thread,  approximately  -    — -; 

p  =  pitch  of  thread; 

P 

a  =  mean  pitch  angle,  i.e.,  tan  a=—  ; 

/*  =  coefficient  of  friction  between  nut  and  thread; 

(/>  =  angle  of  friction  between  nut  and  thread,  i.e.}  tan  (/>  =  JJL. 

ist.  To  raise  W. 

Consider  W  as  a  free  body  moving  uniformly  up  the  incline 
under  the  system  of  forces  shown  in  Fig.  80,  where  W  =  axial 


\ 


FIG.  80. 


load,  R  the  normal  reaction  between  nut  and  thread  due  to  W, 
H  the  horizontal  push  forcing  the  nut  up  the  incline,  and  F 
the  friction  in  direction  of  incline  due  to  the  normal  pressure  R', 
then,  by  the  ordinary  laws  of  mechanics, 


^-cos  a 


(i) 
(2) 
(3) 


The  turning  moment  M 


(4) 


144 


MACHINE  DESIGN. 


Since  tan  a  =  — ,  and  tan  <j>  =  /*,  while 


tan  a  +  tan  <£ 
i— tan  a  tan  6' 


it  follows  that 


2d.  To  lower  W, 


H  =  W 


(5) 

(6) 
(7) 


The  foregoing  has  applied  to  square  threads.  Consider  V 
threads  with  £1  =  the  angle  of  V  with  a  plane  normal  to  the  axis 
of  the  screw.  (See  Fig.  81.)  The  mean  helix  angle = a  as  be- 


FIG.  81. 


fore,  but  now  R  slopes  from  W  in  two  directions,  making  the 
angle  a  in  the  one,  as  before,  but  also  making  the  angle  ft  with 
W  in  a  plane  at  right  angles  with  the  first.  Hence 


R  =  W  sec  a  sec  fa 
F  =  fjtW  sec  a  sec  ft. 


For  raising  load,  approximately, 

p  +  2nr[j.  sec  ft 


Hr  =  Wr1 


(8) 
(9) 

(10) 


BOLTS  AND  SCREH/S. 


145 


For  lowering  load,  approximately, 

—  27T/7*  sec  /? 


Hr  =  Wr- 


sec 


(n) 


88.  Calculation  for  Screws  Not  Stressed  in  Screwing  Up. — 

Returning  now  to  (a).  As  illustrations  of  this  class  consider 
the  eye-bolts  shown  in  Fig.  82.  It  is  customary  to  neglect  the 
influence  of  the  thread  on  the  strength  of  the  bolt,  and  to  con- 
sider as  the  effective  area,  A,  to  resist  stress,  only  the  area  of  a 
circle  whose  diameter  equals  the  diameter  of  the  bolt  at  the 
bottom  of  the  thread.  In  both  cases  considered  a  torsional 
stress  is  induced  by  screwing  the  engaging  surfaces  together, 
but  if  these  surfaces  are  a  proper  fit,  this  stress  is  negligible, 
particularly  since  it  exists  only  within  the  limits  of  the  engaging 
threads  where  the  action  of  the  further  working  load  which  the 
bolt  bears  does  not  come  into  play. 


FIG.  82. 


The  eye-bolt  being  now  subjected  to  the  working  load  T  in 
the  direction  of  its  axis,  we  have 


- 

' 

where  /  is  the  safe  unit  working  stress  for  the  material  and  con- 
ditions. 

If  U  =  ultimate  unit  strength  of  the  material,  then,  if  the 


146  MACHINE  DESIGN. 

load  is  a  constant,  dead  load,  /  may  be  taken  as  great  as  —  for 

o 

good  wrought  iron  or  mild  steel.  If  the  load  is  a  variable  one, 
slowly  applied  and  removed,  /  should  not  exceed  —  for  the 
same  materials.  If  the  load  is  variable  and  suddenly  applied, 
/  should  never  exceed  -  -  for  these  materials,  and  in  cases  of 

shock  may  need  to  be  much  smaller  than  this. 

The  case  shown  in  Fig.  83  must  not  be  confused  with  the 
preceding.  Here  (as  will  be  explained  under  (b) )  there  may  be 
a  tensile  stress  in  A  induced  by  compressing  B-B  between  the 
shoulder  and  the  nut.  If  the  extension  in  the  part  A  of  the 
bolt  due  to  the  application  of  the  force  T  later  be  greater  than 
the  compression  caused  in  B-B  by  the  tightening  of  the  nut, 
then  the  shoulder  will  leave  B-B,  and  simple  tension  =  T  results 
in  all  sections  of  the  bolt  below  the  nut,  as  in  case  (a). 

On  the  other  hand,  if  the  extension  in  the  part  A  of  the 
bolt  due  to  the  subsequent  application  of  T  be  not  so  great  as 
the  original  compression  of  B-B  due  to  the  screwing  up,  then 
we  have  in  the  part  A  a  resultant  tension  greater  than  T.  This 
case  would  come  under  (b). 

89.  Calculation  of  Screws  Stressed  in  Screwing  Up.  — (b) 
Combined  tension  and  torsion  are  induced  in  a  bolt  by  tighten- 
ing it.  The  stress  may  equal,  or  very  greatly  exceed,  the  tensile 
stress  due  to  working  forces.  Consider  the  example  shown  in 
Fig.  78.  Suppose  the  nut  screwed  down  so  that  the  parts  con- 
nected by  the  bolt  are  held  close  together  at  E-F  but  have  not 
yet  been  compressed.  Suppose  that  the  proportions  are  such 
that  the  wrench  may  be  given  another  complete  turn.  The  nut 
will  move  along  the  direction  of  axis  of  the  bolt  a  distance  =  p. 
The  parts  held  between  the  head  and  nut  will  be  compressed 
and  the  body  of  the  bolt  will  be  extended. 


BOLTS  AND  SCREWS.  147 

The  force  applied  at  the  point  B,  or  end  of  the  wrench 
(a  distance  /  from  the  axis)  will  range  from  a  value  of  o  at  the 
beginning  of  the  turn  to  a  value  P  at  the  finish.  The  average 

P 
value  of  the  turning  force  will  be  approximately  =  —. 

The  distance  moved  through  by  the  point  of  application  of 
this  force  is  2x1.  Hence  the  work  done  in  turning  the  nut  a 
full  turn  under  these  conditions  will  be 


-.27il  =  Pnl (12) 


The  resistances  overcome  by  this  application  of  energy  are 
three  in  number: 

ist.  The  work  done  in  extending  the  bolt. 

2d.  The  work  done  in  overcoming  the  frictional  resistance 
between  nut  and  thread. 

3d.  The  work  done  in  overcoming  the  frictional  resistance 
between  nut  and  washer. 

These  will  be  considered  in  order. 

ist.  Let  T  =  the  final  pure  tensile  stress  in  the  bolt  due  to 
screwing  up  one  turn.  At  the  beginning  of  the  turn  the  ten- 

T 

sion  =  o.     The   average   value   may  be  considered  =  —  for  the 

turn.  The  distance  moved  through  by  the  point  of  application 
of  this  force  in  the  direction  of  its  line  of  action,  in  one  turn  =  p. 
The  work  done  in  extending  the  bolt 


(13) 


2d.  The  frictional  resistance  between  the  threads  of  nut  and 
bolt  depends  upon  the  form  of  the  thread  as  well  as  the  mate- 
rials used  and  the  condition  of  the  surfaces.  (See  equations  (2) 


148  MACHINE  DESIGN. 

and  (9),  §  87.)     Assuming  a  V  thread  as  being  more  commonly 
used  for  fastenings,  the  average  value  of  the  friction 

r 

F  =  u—  sec  a  sec  B 

r  2 

T 

(eq.  (9) ),  since  the  average  load  for  the  turn  =  — . 

The  distance  moved  through  by  the  point  of  application  of 
F  for  one  turn  of  the  nut  on  the  bolt  =  ^>  cosec  a.  Hence  the 
work  done  in  overcoming  the  friction  between  bolt  and  nut  in 

T 
the  one  turn  =  /*—  sec  a  sec  /?  .  p  cosec  a 


T 

—  fip  sec  a  sec  /?  .  cosec  a (14) 


3d.  The  frictional  resistance  between  nut  and  washer  due  to 

T  T 

a  mean  force  —  will  be  // — ,  in  which  //  is  the  coefficient  of 

friction  between  nut  and  washer.  The  point  of  application  of 
this  resistance  may  be  taken  at  a  distance  of  —  r\  from  the  axis 

of  the  bolt,  r\  being  the  outside  radius  of  bolt-thread.  The  dis- 
tance moved  through  by  the  point  of  application  for  one  turn  of 
the  nut  =  27rifi,  and  the  work  done  in  overcoming  this  frictional 
resistance 


Equating  (12)  to  the  sum  of  (13),  (14),  and  (15),  gives 


T       T  T  , 

Pnl=—p+—up  sec  a  sec  ft  cosec  a-f— // 
2  r      2  ^r  2 


BOLTS   AND  SCREWS.  149 

whence 


T=  _  __ 

sec  a  sec  ?  cosec  ' 


T 

!t=-7  =  unit  stress  in  bolt  due  to  pure  tension.  .     (17) 

In  addition  to  this  it  must  be  borne  in  mind  that  the  screw 
is  subjected  to  a  torsional  moment  whose  value  can  be  deter- 
mined by  considering  the  nut  as  a  free  body  as 
shown  in  plan  view  in  Fig.  84,  where  all  of  the 
forces  capable  of  producing  moments  about  the 
axis  of  the  nut  are  indicated  as  they  exist  at  the 
end  of  the  turn. 

Summing  the  moments  about  the  axis  of  the 
bolt  gives 

FIG.  84.  Hr  =  Pl-y!T^  .....     (18) 


Hr  is,  of  course,  the  torsional  moment  transmitted  from  the  nut 
to  the  bolt.  To  find  its  numerical  value  substitute  the  value  of 
T  found  in  equation  (16)  and  solve  (18). 

The  unit  stress  induced  in  the  outer  fibers  of  a  rod  of  cir- 
cular section  and  radius  r2  (  =  radius  at  bottom  of  thread)  is 
found  by  means  of  the  equation 


(19) 


J  is  the  polar  moment  of  inertia,  in  this  case  =  -  ;  c  is  the  dis- 

tance from  neutral  axis  to  most  strained  fiber,  in  this  case  =  7*2; 
fa  is  the  induced  unit  stress  in  outer  fiber;  M  is  the  moment,  in 
this  case  =  -H"r.  Combining  equations  (18)  and  (19)  and  sub- 
stituting these  values  gives 


150  MACHINE  DESIGN. 

The  equivalent  tensile  unit  stress  due  to  the  combined  action 
of  ft  and  }8  is  found  from  the  equation  for  combined  tension 
and  torsion, 


/  =  o.35/<  +  0.65V/;2 +  4/*2 (21) 

90.  Problem. — What  is  the  unit  fiber  stress  induced  in  a 
II.  S.  standard  J-inch  bolt  in  screwing  up  the  nut  with  a  pull  of 
one  pound  at  the  end  of  a  wrench  8  inches  long  ?  Arrangement 
of  parts  as  shown  in  Fig.  78. 

In  this  case  di  =  .500  in.,        r\  =  .25  in., 

J2  = -400  in.,        r2  = -2  in., 
^  =  .225  in.,        A  =  .126  sq.  in., 
^-.077  in., 
P-./-O.X5, 

P 

a  =  angle  whose  tangent  is  —  =3°  7', 

sec  OL  —  1.0015,          cosec  a  =  18.39, 
0  =  30°,  sec/?- 1. 155, 

P  =  ilb.,         and          /  =  8  ins. 

From  equation  (16) 

2XiX7rX8 

~. 077 +  . 077  X  0.15X1.0015X1.155X18.39 +  0.15X3X^X0.25 
=  74.467  Ibs. 

From  equation  (17), 


From  equation  (20), 

2(1X8-0.15X74.467x1X0.25) 
/«  —  =* 

=303  Ibs. 


BOLTS  AND  SCREWS.  151 

From  equation  (21), 

7=0.35  X59i  +0.65^  59i2  +  4X3032 
=  757  Ibs. 

If  a  pull  of  one  pound  on  an  8-inch  wrench  applied  to  a  \- 
inch  bolt  can  induce  a  unit  fiber  stress  of  757  Ibs.,  since  equa- 
tions (16)  and  (20)  show  that  the  stress  increases  directly  as  the 
pull,  it  follows  that  a  pull  of  30  Ibs.,  such  as  is  readily  exerted 
by  a  workman,  will  induce  a  stress  of  30X757  =  22,710  Ibs.  per 
square  inch. 

91.  Wrench  Pull. — If  this  turning  up  be  gradual  and  the 
bolt  is  not  subjected  to  working  stresses,  this  would  be  safe  for 
either  wrought  iron  or  mild  steel.  On  the  other  hand,  if  the 
final  turning  be  done  suddenly  by  means  of  a  jerking  motion  or 
a  blow,  or  a  long  wrench  be  used,  or  even  an  extra-strong  grad- 
ual pull  be  exerted,  there  is  evident  danger  of  /  having  a  value 
beyond  the  elastic  limit  of  the  material,  even  reaching  the  ulti- 
mate strength. 

It  will  be  noticed  also  that  the  torsional  action  increases  the 

fiber  stress  over  that  due  to  pure  tension  in  the  ratio  of  — ,  i.e., 

in  this  problem,  an  increase  of  over  25  per  cent.  In  general  this 
increase  will  be  from  15  to  20  per  cent,  depending  chiefly  upon 
the  relation  existing  between  jj.  and  // '.  •  It  should  also  be  noted 
that  the  pure  tension,  T,  induced  in  the  bolt  by  the  moment  PI 
may  be  taken  as  the  measure  of  the  pressure  existing  between 
the  surfaces  E-F  (Fig.  78).  In  our  problem  this  pressure,  for 
P=3o  Ibs.,  would  become  30X74.467=2234  Ibs. 

As  a  general  rule  the  length  of  wrench  used  by  the  workman 
is  fifteen  or  sixteen  times  d\,  the  diameter  of  bolt,  and  it  may  be 
stated  that  T  =  fj<P  for  U.  S.  standard  threads. 


152 


MACHINE   DESIGN. 


92.  Calculation  of   Bolts   Subject  to  Elongation.  —  Next 
sider  the  case  shown  in  Fig.  85.     Suppose  that  the  nut  is  screwed 
up  with  a  resulting  tensile  stress  in  the  bolt 
=  T.     A  working  force  Q  tends  to  separate 
the  bodies  A  and  B  at  C-D.     Assume  that  Q 
acts  axially  along  the  bolt.    The  question   is, 
What  value  may  Q  have  without  opening  the 
joint  C-D? 

A    is  the  cross-sectional  area  of  the  bolt; 

L  is  the  original  length  between  bolt-head 
and  nut  when  A  and  B  are  just  in 
contact  at  C-D  but  not  compressed  ; 


>t 


FIG.  85. 


T0  is  the  tensile  stress  in  bolt  due  to  sere  wing  up  ; 
A     is  the  total  elongation  of  bolt  due  to  TO; 
E   is  the  coefficient  of  elasticity  of  the  bolt  material. 
unit  strain     i 


(i) 


In  any  given  case  this  can  be  solved  for  X. 
Let  A'  be  the  area  of  A  and  B  compressed  by  the  bolt  action; 
X  is  the  total  compression  (i.e.,  shortening)  of  A  and  B, 

due  to  the  tightening  of  the  bolt; 
Co  is  the  total  compressive  stress  which  produces  A'; 

.*.  C0  =  TQ. 
E!  is  the  coefficient  of  elasticity  of  the  material  A,  B.     Then 


L_     2_ 
Co  ~Er 

A' 


(2) 


BOLTS  /tND  SCREWS.  1 53 

This  can  be  solved  for  X' ' . 

Now  consider  the  condition  when  a  working  force,  Q,  acts 
tending  to  elongate  the  bolt  so  that  A  and  B  will  just  be  ready 
to  separate  at  C-D.  In  order  that  this  separation  may  begin, 
the  bolt,  already  elongated  an  amount  ^,  must  be  elongated  a 
further  amount  A'. 

For  incipient  separation  the  total  elongation  of  the  bolt  then 
=  X+Xf,  and  the  total  stress  in  the  bolt  corresponding  to  this 
elongation,  =  T',  can  be  determined  from  the  equation 


Considering  the  bolt-head  as  a  free  body  (Fig.  86),  it  follows 
that  the  forces   acting  on   it  at  any  instant  will  be,  C, 
the  reaction  of  the   material  of  A  due  to  its  resistance 
to  compression;  Q,  the  working  force;  and  T,  the  ten- 
FIG.  86.   sion  in  the  bolt.     Hence 


(4) 


When  the  bolt  is  first  screwed  up,  and  Q=o,  then  C  =  T, 
and  T  =  To,  the  tension  due  to  screwing  up.  When  Q  comes 
into  action,  C  is  partly  relieved,  and  when  Q  reaches  such  a 
value  that  the  surfaces  are  about  to  part,  then  C  =  o  and  Q  = 
T  =  T'.  (See  equation  (3).) 

An  examination  of  these  formulae  shows  certain  facts  which 
may  be  stated  as  follows:  The  tightness  of  the  joint  C-D  de- 
pends upon  the  compressibility  of  A  and  B. 

Anything  which  increases  the  total  compression,  Xf,  increases 
the  tightness  of  the  joint.  This  may  be  accomplished  by  in- 
creasing L  or  Co,  or  decreasing  A'.  It  may  also  be  increased 


154 


MACHINE  DESIGN. 


by  the  introduction  of  a  highly  elastic  body  (i.e.,  gasket)  between 
A  and  B. 

It  also  follows  that  the  tension  in  the  bolt  when  the  joint  is 
about  to  open,  T1',  must  be  greater  than  the  tension  due  to 
screwing  up,  T0,  and  therefore  if  Q  be  limited  to  a  value  equal 
to  or  less  than  TO,  there  will  be  no  opening  of  the  joint.  In 
general,  A'  is  large  compared  with  A,  and  Xf  very  small  com- 
pared with  X,  so  that  T'  is  not  much  greater  than  TQ.  In  order 
to  be  sure  of  a  tight  joint  the  initial  tension  should  be  taken 


93.  Problem  I.  —  Calculate  the  bolts  for  a  "blank"  end  for 
a  6-inch  pipe  using  flanged  couplings  with  ground  joints,  and 
no  gaskets,  as  shown  in  Fig.  87.  The  excess  internal  pressure 
is  to  be  150  Ibs.  per  square  inch. 


FIG.  87. 
The  area  subjected  to  pressure  has  a  diameter  of  7^  ins.; 

2 

hence  the  total  working  pressure  =  i5oX : — =6627  Ibs. 

4 

The  number  of  bolts  is  determined  by  the  distance  they  may 
be  spaced  apart  without  danger  of  leakage  due  to  the  springing 
of  the  flange  between  the  bolts.  This  distance  may  be  taken 
equal  to  four  or  five  times  the  thickness  of  the  flange.  In  the 
problem  under  consideration,  the  diameter  of  the  bolt  circle  will 
be  approximately  9  ins.,  and  using  six  bolts,  the  chord  length 
between  consecutive  ones  will  be  about  4^  ins.,  which  is  per- 
fectly safe. 


BOLTS  AND  SCREWS.  155 

With  six  bolts 


Take  T0  =  2Q  =  2210  Ibs. 

With  a  direct  tension  of  2210  Ibs.  due  to  screwing  up,  there 
is  also  the  stress  due  to  torsion.  As  stated  in  §  91,  this  may 
increase  the  fiber  stress  20  per  cent  over  that  due  to  direct  ten- 
sion. To  allow  for  this  the  bolts  used  must  be  capable  of  safely 
sustaining  a  stress  of  2210X1.20  =  2650  Ibs. 

The  allowable  unit  stress  here  may  be  taken  rather  high, 
since  the  conditions  after  once  screwing  up  approximate  a  steady 
load.  Assume  steel  bolts  with  an  allowable  unit  stress  of 
15,000  Ibs. 

The  area  of  each  bolt  at  the  bottom  of  the  thread  will  then 

be  -  =0.177  scl-  m-      This  value  lies  between  a  &-inch  and 
15000 

a  f-inch  bolt.  Select  the  latter  with  an  area  of  0.202  sq.  in.  To 
exert  an  initial  tension  of  2250  Ibs.  in  a  f-inch  bolt  would  re- 
quire a  pull  of  about  30  Ibs.  on  a  lo-inch  wrench.  (See  §  91.) 
These  values  just  about  correspond  to  actual  conditions  in  prac- 
tice. 

94.  Problem  II.  —  It  is  required  to  design  the  fastenings  to 
hold  on  the  steam-chest  cover  of  a  steam-engine.  The  opening 
to  be  covered  is  rectangular,  io"Xi2".  The  maximum  steam- 
pressure  is  100  Ibs.  per  square  inch.  The  joint  must  be  held 
steam-tight.  Studs  of  machinery  steel  having  an  ultimate  ten- 
sile strength  of  60,000  Ibs.  per  square  inch  will  be  used. 

The  total  working  pressure  =  loX  12  X  100=  12,000  Ibs. 

The  number  of  studs  to  be  used  will  be  governed  by  the  dis- 
tance they  may  be  spaced  without  springing  of  the  cover.  The 
thickness  of  the  latter  being  assumed  to  be  f  inch  at  the  edge, 
the  spacing  should  not  exceed  5  Xf"  =  1/",  say  4  inches. 


156 


MACHINE  DESIGN. 


The  opening  is  io"Xi2",  as  shown  in  Fig. 
be  a  band  about  f  or  f  inch  wide 
around  this  for  making  the  joint, 
upon  which  the  studs  must  not  en- 
croach. This  makes  the  distance 
between  the  centers  of  the  vertical 
rows  of  studs  about  14  inches,  and 
between  the  horizontal  rows  about 
12  inches.  Twelve  studs  can  be  used 
if  arranged  as  shown  in  the  figure. 
The  greatest  distance,  that  between 
the  studs  across  the  corners,  will  but 
slightly  exceed  the  allowable  4  inches. 

With  12  studs,  the  working  load  on  each  =  Q  = 


O 


There  must 


O 


J 

I 

O          O 
FIG.  88. 

12000 
12 


1000  Ibs. 


TQ  =  2Q  =  2000  Ibs. 

Allowing  20  per  cent  for  torsional  stress,  increases  this  to 
2400  Ibs. 

Allowing  a  unit  stress  of  15,000  Ibs.,  as  in  Problem  I,  we 

have  as  the  area  of  the  stud  at  the  bottom  of  thread  -     —  =  o.  160. 

15000 

This  corresponds  to  a  A-inch  stud.  Since  a  workman  may 
readily  stress  a  bolt  of  this  size  beyond  the  elastic  limit  by  exert- 
ing too  great  a  pull  in  tightening,  many  designers  would  increase 
these  studs  to  f  inch  or  even  f  inch. 

95.  Design  of  Bolts  for  Shock. — The  elongation  of  a  bolt 
with  a  given  total  stress  depends  upon  the  LENGTH  and  AREA 
of  its  least  cross-section.  Suppose,  to  illustrate,  that  the  bolt, 
Fig.  89,  has  a  reduced  section  over  a  length  /  as  shown.  This 
portion,  A,  has  less  cross-sectional  area  than  the  rest  of  the  bolt, 
and  when  any  tensile  force  is  applied,  the  resulting  UNIT  stress 
will  be  greater  in  A  than  elsewhere.  The  unit  strain,  or  elonga- 
tion, will  be  proportionately  greater  up  to  the  elastic  limit;  and 


BOLTS  AND  SCREWS. 


157 


if  the  elastic  limit  is  exceeded  in  the  portion  A,  the  elongation 
there  will  be  far  greater  than  elsewhere.  If  there  is  much  differ- 
ence of  area  and  the  bolt  is  tested  to  rup- 
ture, the  elongation  will  be  chiefly  at  A. 
There  would  be  a  certain  elongation  PER 
INCH  of  A  at  rupture.  Hence  the  greater 
the  length  of  A,  the  greater  the  total  elonga- 
tion of  the  bolt.  If  the  bolt  had  not  been 
reduced  at  A,  the  minimum  section  would 
be  at  the  root  of  the  screw-threads.  The 
FIG.  89.  FIG.  90.  axiai  lcngth  of  this  section  is  very  small. 
Hence  the  elongation  at  rupture  would  be  small.  Suppose  there 
are  two  bolts,  A  with  and  B  without  the  reduced  section.  They 
are  alike  in  other  respects.  They  are  subjected  to  equal  tensile 
shocks.  Let  the  energy  of  the  shock  =  E.  This  energy  is  di- 
vided into  force  and  space  factors  by  the  resistance  of  the  bolts. 
The  space  factor  equals  the  elongation  of  the  bolt.  This  is 
greater  in  A  than  in  B,  because  of  the  yielding  of  the  reduced 
section.  But  the  product  of  force  and  space  factors  is  the  same 
in  both  bolts,  =E;  hence  the  resulting  stress  in  the  minimum 
section  is  less  for  A  than  for  B.  The  stress  in  A  may  be  less 
than  the  breaking  stress,  while  the  greater  stress  in  B  may 
break  it.  THE  CAPACITY  OF  THE  BOLT  TO  RESIST  SHOCK  is 

THEREFORE    INCREASED     BY     LENGTHENING     ITS    MINIMUM     SEC- 
TION TO  INCREASE   THE   YIELDING  AND  REDUCE  STRESS.      This  IS 

not  only  true  of  bolts,  but  of  all  stress  members  in  machines. 

The  whole  body  of  the  bolt  might  have  been  reduced,  as 
shown  by  the  dotted  lines  in  Fig.  89,  with  resulting  increase  of 
capacity  to  resist  shock.  Turning  down  a  bolt,  however,  weak- 
ens it  to  resist  torsion  and  flexure,  because  it  takes  off  the  material 
which  is  most  effective  in  producing  large  polar  and  rectangular 
moments  of  inertia  of  cross-section.  If  the  cross-sectional  area 
is  reduced  by  drilling  a  hole,  as  shown  in  Fig.  90,  the  torsional 


158  MACHINE  DESIGN. 

and  transverse  strength  is  but  slightly  decreased,  but  the  elon- 
gation will  be  as  great  with  the  same  area  as  if  the  area  had 
been  reduced  by  turning  down. 

Professor  Sweet  had  a  set  of  bolts  prepared  for  special  test. 
The  bolts  were  ij  inches  diameter  and  about  12  inches  long. 
They  were  made  of  high-grade  wrought  iron,  and  were  dupli- 
cates of  the  bolts  used  at  the  crank  end  of  the  connecting-rod 
of  one  of  the  standard  sizes  of  the  Straight-line  Engine.  Half 
of  the  bolts  were  left  solid,  while  the  other  half  were  carefully 
drilled  to  give  them  uniform  cross-sectional  area  throughout. 
The  tests  were  made  under  the  direction  of  Professor  Carpenter 
at  the  Sibley  College  Laboratory.  One  pair  of  bolts  was  tested 
to  rupture  by  tensile  force  gradually  applied.  The  undrilled 
bolt  broke  in  the  thread  with  a  total  elongation  of  0.25  inch. 
The  drilled  bolt  broke  between  the  thread  and  the  bolt-head 
with  a  total  elongation  of  2.25  inches.  If  it  be  assumed  that 
the  mean  force  applied  was  the  same  in  both  cases,  it  follows 
that  the  total  resilience  of  the  drilled  bolt  was  nine  times  as  great 
as  that  of  the  solid  one.  "Drop  tests,"  i.e.,  tests  which  brought 
tensile  shock  to  bear  upon  the  bolts,  were  made  on  other  similar 
pairs  of  bolts,  which  tended  to  confirm  the  general  conclusion. 

96.  Problem. — It  is  required  to  design  proper  fastenings  for 
holding  on  the  cap  of  a  connecting-rod  like  that  shown  in  Fig. 
91.  These  fastenings  are  required  to  sus- 
tain shocks,  and  may  be  subjected  to  a 
maximum  accidental  stress  of  20,000  Ibs. 
There  are  two  fastenings,  and  therefore 
each  must  be  capable  of  sustaining  safely 
a  stress  of  10,000  Ibs.  They  should  be 


designed  to  yield  as  much  as  is  consistent 


TTUN          /tut 


with  strength ;  in  other  words,  they  should 
be  tensile  springs  to  cushion  shocks,  and 
thereby  reduce  the  resulting  force  they  have  to  sustain.  Bolts 


BOLTS  4ND  SCREWS.  159 

should  therefore  be  used,  and  the  'weakest  section  should  be 
made  as  long  as  possible.  Wrought  iron  will  be  used  whose 
tensile  strength  is  50,000  Ibs.  per  square  inch.  The  stress  given 
is  the  maximum  accidental  stress,  and  is  four  times  the  working 
stress.  It  is  not,  therefore,  necessary  to  give  the  bolts  great 
excess  of  strength  over  that  necessary  to  resist  actual  rupture 
by  the  accidental  force.  Let  the  factor  of  safety  be  2.  This 
will  keep  the  maximum  fiber  stress  within  the  elastic  limit. 
Then  the  cross-sectional  area  of  each  bolt  must  be  such  that 
it  will  just  sustain  10,000X2=20,000  Ibs.  This  area  is  equal 
to  20,000-^50,000=0.4  sq.  in.  This  area  corresponds  to  a 
diameter  of  0.71  inch,  and  that  is  nearly  the  diameter  of  a 
f-inch  bolt  at  the  bottom  of  the  thread;  hence  J-inch  bolts  will 
be  used.  The  cross-sectional  area  of  the  body  of  the  bolt 
must  now  be  made  at  least  as  small  as  that  at  the  bottom  of  the 
thread.  This  may  be  accomplished  by  drilling. 

97.  Jam-nuts. — When  bolts  are  subjected  to  constant  vibra- 
tion there  is  a  tendency  for  the  nuts  to  loosen.  There  are  many 
ways  to  prevent  this,  but  the  most  common  one  is  by  the  use  of 
jam-nuts.  Two  nuts  are  screwed  on  the  bolt;  the  under  one 
is  set  up  against  the  surface  of  the  part  to  be  held  in  place,  and 
then  while  this  nut  is  held  with  a  wrench  the  other  nut  is  screwed 
up  against  it  tightly.  Suppose  that  the  bolt  has  its  axis  vertical 
and  that  the  nuts  are  screwed  on  the  upper  end.  The  nuts  being 
screwed  against  each  other,  the  upper  one  has  its  internal  screw 
surfaces  forced  against  the  under  screw  surfaces  of  the  bolt,  and 
if  there  is  any  lost  motion,  as  there  almost  always  is,  there  will 
be  no  contact  between  the  upper  surfaces  of  the  screw  on  the 
bolt  and  the  threads  of  the  nut.  Just  the  reverse  is  true  of  the 
under  nut;  i.e.,  there  is  no  contact  between  the  under  surfaces 
of  the  threads  on  the  bolt  and  the  threads  on  the  nut.  There- 
fore no  pressure  that  comes  from  the  under  side  of  the  under 
hut  can  be  communicated  to  the  bolt  through  the  under  nut 


i6o 


MACHINE  DESIGN. 


directly,  but  it  must  be  received  by  the  upper  nut  and  com- 
municated by  it  to  the  bolt,  since  it  is  the  upper  nut  alone  that 
has  contact  with  the  under  surfaces  of  the  thread.  Therefore 
the  jam-nut,  which  is  usually  made  about  half  as  thick  as  the 
other,  should  always  be  put  on  next  to  the  surface  of  the  piece 
to  be  held  in  place.  Other  locking  devices  are  shown  in  Fig. 


FIG.  91.4. 

98.  Calculation    of    Screws   for   Transmission   of   Power. — 

(c)  Screws  are  frequently  used  to  transmit  power.  A  screw-press  is 
a  common  exam  pie,  while  the  action  of  spiral  gears,  including  worms 
and  wheels,  is  that  of  screws  and  subject  to  the  same  analysis. 
Collar  friction,  or  nut  and  washer  friction,  is  here  neglected. 
The  use  of  ball  or  roller  thrust  bearings  permits  this.  Equation 
(18)  shows  how  it  may  be  introduced  if  PI  does  not  equal  Hr. 

The  general  action  of  screw  and  nut  has  already  been  treated. 
(See  §  87.) 

With  a  square-thread  screw  it  'was  found  that  the    moment 

Pl  =  M,  required  to  raise  a  load  W,  will  ^e=^r2nr_^  (S) 
(page  144). 

This  will  induce  a  fiber  stress  /«=~~F~  (see  equation  (19)  ), 


BOLTS   AND   SCREWS. 


161 


w 


which  must  be  combined  with  the  tension,  }t  =-£,  in  order  to 
get  the  actual  unit  stress,  /,  remembering 


(21) 


Let  w=number  of  complete  thread  surfaces  in  contact 
between  the  nut  and  screw,  and  the  projected  area  equals 

n-(di2-d22)  to  bear  the  load  W. 
4 

W=Kn-(d12-d2*), 
4 

where  K  is  the  allowable  pressure  per  square  inch  of  projected 
thread  area. 

For  nuts  and  bolts  which  are  used  as  fastenings  we  may  take: 
^  =  2500  Ibs.  for  wrought  or  cast  iron  running  on  the  same 

material  or  on  bronze  ; 
^  =  3000  Ibs.  for  steel  on  steel  or  bronze. 

With  good  lubrication,  where  the  screw  and  nut  are  used  to 
transmit  power,  we  may  take  the  values  given  in  the  following 
table  : 

TABLE  VII. 


Rubbing  Speed  in  Feet 
per  Minute. 

Value  of  K. 

Iron. 

Steel. 

CQ  or  less  

2500  + 
1250 
850 
400 

200 

3000  + 
1500 

1000 

500 
250 

100  

I  C.O 

*y*'  • 

2  C.O 

4OO 

The  value  of  //  has  been  experimentally  determined  by  Pro- 
fessor Kingsbury.*     He  concludes  that  for  metallic  screws  turn- 


*  Transactions  A.  S.  M.  E.,  Vol.  XVII,  pp.  96-116. 


162 


MACHINE  DESIGN. 


ing  at  extremely  slow  speeds,  under  any  pressure  up  to  14,000 
Ibs.  per  square  inch  of  bearing  surface,  and  freely  lubricated 
before  application  of  the  pressure,  the  following  coefficients  of 
friction  may  be  used. 


TABLE  VIII. 


Lubricant. 

f. 

Lard-oil             ....          .  .      . 

Heavy-machinery  oil  (mineral)    .... 

O    143 

Heavy-machinery  oil  and  graphite  in  equal 
volumes  

o  o7 

Regarding  the  efficiency  of  the  square-screw  thread  to  trans- 
mit energy,  we  may  reason  as  follows : 

useful  work 

The  efficiency  =e  =  ~~  — ; :-, 

J  total  work 


which  becomes  for  one  turn  = 


Wp 
2nrH' 


Wp  271 

2nrH~    H 


TFtan 


tan  a 


tan 


.  .     .     (22) 


From  this  it  appears  that  e  becomes  o,  for  a=o°  and  for 
^  =  90°  —  <£,  and  must  therefore  have  a  maximum  value  between 
these  limits.  To  determine  this  maximum,  write 


tan  a 


tan 


=  tan  a  cot  (a 


Taking  the  first  differential  and  equating  to  o, 
cot  (a +  6)  tan  a 




cos2  a          sin2  (a  +  0) 


=  o. 


Solving  which 'gives 


BOLTS   AND  SCREWS.  163 

To  find  the  corresponding  value  of  e,  write  (from  (22)  ) 


max.  e 


tan(45°-f)        tan(45°-~) 


To  lower  W  with  a  square-  threaded  screw, 


[The  value  of  }8  can  be  found  from  this,  as  explained  in  the 
section  on  raising  W,  and  combined  with  jt  to  obtain  /.] 
Regarding  the  efficiency  in  this  case, 

if  a<  <j>  the  load  will  not  sink  of  itself  (i.e.,  overhaul), 

if  a  =  (j>  we  have  a  condition  of  equilibrium, 

if  a  ></>  the  load  will  sink  of  itself  (i.e.,  overhaul). 

For  a  screw  which  will  not  overhaul  it  becomes  evident  that 
the  limiting  value  of  a  is  $  and  the  maximum  efficiency 

tan  a  tan  d>      i  —tan2  d> 

=  —          —=-0.5—  0.5  tan2  (/>. 


tan    «  +  <)     tan 


The  efficiency  of  a  screw  which  will  not  overhaul  can  there- 
fore never  exceed  0.5  or  50  per  cent. 

For  V  threads,  with  ,5  =  angle  of  V,  with  a  plane  normal  to 
the  axis  of  the  screw  for  raising  load, 


2nr-pfj.  sec  /? 
which  is  evidently  greater  than  (5),  and 

tan  a  (  i  —  /*  tan  a  sec  /?) 
tan  a  +     sec 


1  64  MACHINE  DESIGN. 

which  is  evidently  less  than  e  for  square  threads.  (See  equa- 
tion (22).) 

It  is  clear,  then,  that  square  threads  should  be  used  in  pref- 
erence to  V  threads  for  screws  for  power  transmission. 

For  lowering  the  load  with  V  threads 


.          (26) 

c  ,3 

99.  Problem.  —  Design  a  screw  to  raise  20,000  Ibs.  The 
screw  must  not  overhaul.  No  collar  friction. 

What  moment  need  be  exerted  to  lift  the  load  ? 

What  will  be  the  efficiency  of  the  screw  ? 

Select  a  square-thread  screw  of  machinery  steel  running  in 
a  bronze  nut. 

For  a  screw  which  will  not  overhaul  a  must  be  less  than  </>. 

.*.  tan  a<jj.. 

To  be  safe  against  overhauling  with  the  materials  used  and 
good  lubrication,  //  must  not  be  given  a  greater  value  than  o.io. 

/.  a<5°45'     and     <j>  =  $°  45'. 

The  pure  tension  =  20,000  lbs.  =  PF.     In  the  preliminary  calcu- 
lations, to  allow  for  the  effect  of  torsion,  this  will  be  increased 

25000 
so  that  /=•      ,     . 

In  this  equation  /  is  the  allowable  unit  stress  in  pounds  per 
square  inch,  and  A  is  the  area  of  the  screw  at  the  bottom  of  the 
thread  in  square  inches. 

Assume  that  this  screw  is  frequently  loaded  and  unloaded, 
and  not  subject  to  shocks  nor  reversal  of  stress  so  that  /  = 

12,000  Ibs.  per  square  inch  for  mild  steel.     Then  A  =  --  = 

12000 


BOLTS   AND  SCREWS  165 

2.08  sq.  ins.     This  corresponds  to  a  diameter  of  if  inches  at 
bottom  of  thread. 

P 

From  tan  a  =  — ,  we  have 
2;rr 

p  =  2nr  tan  a. 

Remembering  that  for  square  threads  the  depth  of  the  thread 
=  -£,  and   that  r2  is  the  radius  at  the   bottom   of   thread,  and 


2 


/.*.  r=r2  +  — ,  it  follows  that 
4 


P 

—  )  tan  a, 

4 


27r  tan  a 
p  --        —  ^ 

tan  a 


i  —  —  tan  a 

2 
2X7rX0.8l25XO.I 

/.  p  =  -  -  =  .606  inch. 

1-1.57X0.1 

This  is  not  a  thread  to  be  easily  cut  in  the  lathe.  It  would 
be  desirable  to  modify  the  value  of  p  so  that  the  thread  can  be 
readily  cut.  It  is  obvious  that  p  cannot  be  increased  without 
increasing  r  proportionately,  else  a  will  have  a  greater  value 
than  is  allowable.  It  will  be  more  economical  to  reduce  p. 
The  nearest  even  value  would  be  J  inch,  and  this  will  be  selected. 
Check  this  for  strength: 

^2  =  1-625  inches,       r2=  0.8125  inch,  p  =  0.5  inch. 

^1  =  2.125  inches,       ri  =  1.0625  inches, 
^=1.875  inches,         r=0-937S  inch, 


i66  MACHINE  DESIGN. 

tan  a=-^-=  -  —  -  =  0.087, 
27rr     2X^X0.9375 

which  is  safe,  as  it  is  less  than  the  value  of  /z  =  o.i 
From  equation  (5),  the  moment, 


=  3496  in.-lbs. 
From  equation  (19),  the  fiber  stress  due  to  torsion 

CPI       2PI 


From  equation  (17),  the  unit  stress  due  to  tension, 

W     20000  .     , 

jt=—  =  -  =  9597  lbs.  per  square  inch. 

From  equation  (21),  the  combined  stress, 


=  12,379  lbs.  per  square  inch, 

which  is  near  enough  12,000  to  be  considered  safe. 
The  efficiency,  from  equation  (22), 


tan  a 
e  = 


tan  (a  +  </>)' 


BOLTS  AND  SCREWS.  167 

Since  tan  a  =0.087,         a  =  5°. 
Since  /JL  =  tan  $  =  o.  10,    <£  =  5°  45'. 

/.  a  -f  <£  =  io°  45',     tan  10°  45'  =  0.1899. 


The  height  of  the  nut  is  determined  from  the  equation 


in  which  W  is  the  load,  n  the  number  of  complete  threads  in 
the  nut,  di  the  outside  and  d2  the  inside  diameter  of  thread. 
K  is  the  allowable  pressure  in  pounds  per  square  inch,  and  its 
value  depends  upon  the  speed.  See  table  in  sec.  98. 

Assuming  the  screw  to  have  a  rubbing  velocity  of  less  than 
50  feet  per  minute,  K  =  3000.     Then 

W  20000 


3°°°Xo.7854(2.i2S*-  1.625*) 


=  4.5,  nearly. 
The  height  of  nut  = 


CHAPTER  IX. 

MEANS   FOR   PREVENTING   RELATIVE  ROTATION. 

100.  Classification  of  Keys. — Keys  are  chiefly  used  to  pre- 
vent relative  rotation  between  shafts  and  the  pulleys,  gears,  etc., 
which  they  support.     Keys  may  be  divided  into  parallel  keys, 
taper  keys,  disk  keys,  and  feathers  or  splines. 

101.  Parallel  Keys. — For  a  PARALLEL  KEY  the  "seat,"  both 
in  the  shaft  and  the  attached  part,  has  parallel  sides,  and  the 
key  simply  prevents  relative  rotary  motion.     Motion  parallel  to 
the  axis  of  the  shaft  must  be  prevented  by  some  other  means, 
as  by  set-screws  which  bear  upon  the  top  surface  of  the  key,  as 
shown  in  Fig.  92.     A  parallel  key  should  fit  accurately  on  the 
sides  and  loosely  at  the  top  and  bottom.     Parallel  keys  may  be 
"  square  "  or  "  flat."     The  following  table  (IX)  for  dimensions 
for  square  keys  is  from  Richards's  "  Manual  of  Machine  Con- 
struction." 

TABLE  IX. 

Diameter  of  shaft  =  d=  i         i\       i|       if       2         i\       3         3i       4 
Width  of  key        =w=   &A&HMMH&H 
Height  of  key       =  /=   A       i         At         A       i         At         f 

Excellent  parallel  keys  are  made  from  cold-rolled  steel  with- 
out need  of  any  machining. 

John  Richards's  rule  for  flat  keys  is  (see  Fig.  93)  w  =  -.     t  has 

4 

such  value  that  a  =  30.°  This  rule  is  deviated  from  somewhat, 
as  shown  by  the  following  table  (X),  taken  from  his  "  Manual 
of  Machine  Construction,"  page  58: 

168 


MEANS   FOR  PREVENTING    RELATIVE  ROTATION.  169 

TABLE  X. 

d=i         ii       i£       if       2         2\      3         3!       4         5         6         7        8 

w=  J        A      I        A      i        I        i        I    .  «        ii      it      ij      if 

/=  &      A      i        A      *      I        As        f        tt      H      1      i 

When  two  or  more  keys  are  used,  w  =  d+6,  t  being,  as  before, 
of  such  value  that  a  shall  =  30°. 

1 02.  Taper  Keys. — A  TAPER  KEY  has  parallel  sides  and  has 
its  top  and  bottom  surfaces  tapered,  and  is  made  to  fit  on  all 


FIG.  92.  FIG.  93.  FIG.  94. 

four  surfaces,  being  driven  tightly  "  home."  It  prevents  rela- 
tive motion  of  any  kind  between  the  parts  connected.  If  a  key 
of  this  kind  has  a  head,  as  shown  in  Fig.  94,  it  is  called  a  "  draw 
key,"  because  it  is  drawn  out  when  necessary  by  driving  a 
wedge  between  the  hub  of  the  attached  part  and  the  head  of 
the  key.  Projecting  draw-heads  are  to  be  avoided  on  all  rotating 
parts  unless  guarded  to  prevent  accident.  When  a  taper  key 
has  no  head  it  is  removed  by  driving  against  the  point  with  a 
"  key-drift." 

The  taper  of  keys  varies  from  J  to  \  inch  to  the  foot. 

103.  Fitting  Shaft  and  Hub.  —  In  using  taper  keys  it  is  cus- 
tomary to  bore  out  the  hub  slightly  larger  than  the  diameter  of 
the  shaft  so  that  the  wheel  may  be  removed  readily  after  the 
key  is  withdrawn.  This  allowance  in  diameter  should  not  be 
greater  than  that  for  a  sliding  fit,  say, 


1000 


MACHINE  DESIGN. 


in  which  formula  A  is  the  difference  in  diameter  between  the 
bore  of  hub  and  size  of  shaft,  expressed  in  decimal  parts  of  an 
inch,  and  D  is  the  nominal  diameter  of  shaft  in  inches.  Where 
the  parts  do  not  have  to  be  taken  apart  frequently,  it  is  vastly 
better  to  use  a  driving  fit,  i.e.,  to  bore  out  the  hub  smaller  than 

D 


the  diameter  of  the  shaft  by  an  amount  A  = 


1000 


and  to   use 


parallel  keys. 

Where  a  single  taper  key  is  used  the  effect  is  to  make  the 
wheel  and  shaft  eccentric,  as  can  be  seen  in  Fig.  95.  The  bear- 
ing is  limited  to  two  points,  A,  B,  and  the  connection  is  unstable 
for  the  transmission  of  power. 


FIG.  95. 


FIG.  96. 


FIG.  97. 


If  great  care  is  not  exercised  in  having  the  taper  of  keyway 
exactly  the  same  as  the  taper  of  the  keys,  a  further  difficulty 
arises  in  that  the  wheel  will  be  canted  out  of  a  true  normal  plane 
to  the  shaft-axis.  This  can  be  seen  in  Fig.  96. 

By  using  two  keys,  placed  a  quarter  or  third  of  the  circum- 
ference apart,  a  much  more  stable  connection  is  obtained,  as  it 
will  give  three  points  of  bearing,  A,  B,  and  C.  <See  Fig.  97.) 
Eccentricity  is  not  avoided  by  this  method. 

104.  Woodruff  Keys. — The  Woodruff  or  disk  system  of  keys 
is  used  by  some  manufacturers.  The  key  is  a  half  disk,  as  can 


MEANS  FOR  PRESENTING  RELATIVE  ROTATION. 


171 


be  seen   in   Fig.  98.     Under  this  system  the  keyway  is   cut  lon- 
gitudinally in  the  shaft  by  means  of  a  milling-cutter.     This  cut- 


FIG.  98. 


FIG.  99. 


ter  corresponds  in  thickness  to  the  key  to  be  inserted,  and  is  of 
a  diameter  corresponding  to  the  length  of  the  key.     The  key 


E 


FIG.  100. 


being  semicircular,  it  is  sunk  into  the  shaft  as  far  as  will  allow 
sufficient  projection  of  the  key  above  the  surface  to  engage  the 
keyway  in  the  hub. 


17*  MACHINE   DESIGN. 

Owing  to  its  peculiar  shape  the  key  may  be  slightly  inclined, 
so  that  it  will  serve  to  support  the  wheel  on  a  vertical  shaft,  pro- 
vided the  key-seat  in  the  hub  is  made  tapering  and  of  the  proper 
depth. 

104.  Saddle,  Flat  and  Angle  Keys. — Saddle  keys  (Fig.  99,  A) 
and  keys  on  flats  (Fig.  99,  B)  are  used  occasionally.  They 
have  not  the  holding  power  of  sunk  keys. 

The  type  of  key  shown  in  Fig.  100  A  has  much  to  be  said  in 
its  favor  both  as  regards  ease  and  accuracy  in  obtaining  a  stable 
connection  and  also  as  regards  suitability  of  form  to  resist  stresses. 
It  will  be  noted  that  the  surfaces  are  normal  to  the  lines  of  action 
of  the  forces  transmitted.  The  pressure  per  square  inch  should 
not  exceed  17,000  Ibs.  The  height  of  key  is  taken  equal  to 
0.2  diameter  of  shaft. 

The  Kernoul  key,  shown  in  Fig.  100  B,  is  for  use  in  driving 
in  only  one  direction.  A  portion  of  the  hub  is  cut  out  so  as  to 
form  an  eccentric  slot.  In  this  the  key  fits  as  shown.  The 
inner  face  of  the  key,  curved  to  the  radius  of  the  shaft,  should 
be  left  rough  so  as  to  seize  the  shaft,  while  the  outer  face,  curved 
to  fit  the  slot  in  the  hub,  is  smooth  finished.  When  the  shaft 
rotates  (in  this  case)  counter-clockwise,  the  resistance  to  the 
hub's  motion  being  then  as  indicated  by  the  arrow,  the  surface 
of  the  slot  tends  to  slide  up  on  the  key,  causing  it  to  wedge  in 
between  the  shaft  and  hub,  forming  a  firm  connection.  When 
the  shaft  rotates  in  the  opposite  direction  and  the  resistance 
to  the  hub's  motion  is  reversed,  the  slot  of  the  hub  tends  to  leave 
the  key,  relieving  the  pressure  and  permitting  easy  removal  from 
the  shaft.  At  "  a  "  and  "b  "  there  are  counter-sunk  screws  for 
setting  up  and  loosening  the  key.  In  Fig.  100  C  is  shown  a  special 
form  of  this  type  of  key.  It  is  known  as  the  Barbour  key  and 
is  chiefly  used  for  fastening  the  cams  on  the  shafts  of  stamp 
mills.* 

*  Patent  held  for  this  purpose  by  the  Risdon  Iron  Works.  The  distinguishing 
feature  from  the  plain  Kernoul  key  lies  in  the  use  of  the  inside  projecting  tongue 
which  fits  in  a  key  way  cut  in  the  shaft 


MEANS  FOR  PRESENTING  RELATIVE  ROTATION.  173 

In  the  study  of  keys  which  drive  in  one  direction  only  it  is 
proper  to  include  the  roller  ratchet.  The  simplest  form  is  shown 
in  Fig.  100  D.  The  hub  is  recessed  as  shown  and  the  roller  R 
placed  in  the  recess,  held  in  position  by  a  spring.  The  direction 
of  shaft  rotation  and  hub  resistance  being  as  shown  the  roller 
becomes  wedged  between  the  two,  forming  a  driving  connection. 
With  reversal  of  direction  the  roller  is  freed  and  the  shaft  and 
hub  may  have  relative  motion. 

Generally  more  than  one  roller  is  used  and  the  mechanism 
takes  the  form  shown  in  Fig.  100  E.  A  is  a  hardened  and 
ground  steel  ring  or  bushing  and  B  should  also  be  hardened. 
Each  roller  should  be  held  in  place  by  a  spring  as  shown  at  C. 
Such  ratchets  permit  of  rapid  reciprocation.  Complete  details 
and  descriptions  of  further  clutches  of  this  type  will  be  found  in 
the  American  Machinist  of  Dec.  21,  1905. 

105.  Strength  of  Sunk  Keys. — The  strength  of  the  latter  is 
the  measure  of  their  holding  power.  A  key  of  width  =w,  thick- 
ness =/,  length  =/,  unit  shearing  strength  =/s,  and  unit  crushing 
resistance  =  }c  will  have  a  shearing  strength  =  jswl  and  a  crush- 
ing resistance  ]c\tl. 

If  r  =  radius  of  the  shaft,  the  moment  which  the  key  can 
resist  will  be  measured  by  rwl}8  or  %rtlfc,  whichever  is  smaller 
in  value. 

All  dimensions  being  expressed  in  inches  and  resistance 
in  pounds  per  square  inch,  the  moments  will,  of  course,  be 
expressed  in  inch-pounds.  Experiments  made  by  Professor 
Lanza  indicate  that  the  ultimate  value  for  f8  for  cast  iron  = 
30,000  Ibs.,  for  wrought  iron  =  40,000  Ibs.,  and  for  machinery 
steel  =  60,000  Ibs.  A  factor  of  safety  of  at  least  2  would  be 
advisable  with  these  values. 

H.  F.  Moore  has  experimented  on  the  effect  of  key-ways 
on  strength  of  shafts.*  He  deduces  the  following  expressions 
for  effect  on  stiffness  and  strength,  respectively: 

*  Bulletin  No.  42,  Univ.  of  Illinois,  Eng.  Exp.  Station. 


1 74  MACHINE  DESIGN. 

k=1.0+0.4iv'  +0.*jh, 
e=i.o  -o.2Wf  -i.ih; 
k  =  ratio  of  angle  of  twist  of  shaft  with  key-way  to  angle  of 

twist  of  similar  shaft  without  key-way. 

e  =  ratio  of  strength  at  elastic  limit  of  shaft  with  key- way  to 
the  strength  at  elastic  limit  of  a  similar  shaft  without 
key-way;  called  efficiency; 
w'  =  width  of  key-way -^  diameter  of  shaft; 
h  =  depth  of  key- way  -f-  diameter  of  shaft. 

The  equations  hold  for  values  of  w'  up  to  0.5,  and  for  h  up 
to  0.1875.  The  efficiency  was  not  affected  by  the  addition  of  a 
bending  moment  as  great  as  the  twisting  moment.  The  efficiency 
of  a  shaft  with  two  key-ways  cut  in  the  same  plane  for  two 
Woodruff  keys,  of  such  size  that  the  strength  of  the  solid  shaft 
was  equal  to  the  shearing  strength  of  the  two  Woodruff  keys, 
was  found  to  be  about  the  same  as  the  efficiency  of  a  shaft 
with  a  key-way  whose  width  was  one-fourth  and  whose  depth 
was  one-eighth  of  the  shaft  diameter,  being  about  85  per  cent 
in  both  cases. 

106.  Feathers  or  Splines  are  keys  that  prevent  relative  rota- 
tion, but  purposely  allow  axial  motion.  They  are  sometimes 


FIG.  101.  FIG.  102. 

made  fast  in  the  shaft,  as  in  Fig.  101,  and  there  is  a  key  "way" 
in  the  attached  part  that  slides  along  the  shaft.  Sometimes  the 
feather  is  fastened  in  the  hub  of  the  attached  part,  as  shown 
in  Fig.  102,  and  slides  in  a  long  keyway  in  the  shaft. 

It  is  frequently  undesirable  to  have  the  feather  loose.     In 
such  cases  it  is  common  to  use  tit-keys  as  shown  in  Fig.  103. 


MEANS  FOR   PREVENTING  RELATIVE  ROTATION. 


175 


The  keys  may  be  fastened  to  either  hub  or  shaft.  The  tits 
are  forged  on  the  keys.  Corresponding  holes  are  drilled 
and  countersunk  in  the  piece  to  which  the  key  is  to  be  fastened, 
and  after  the  key  is  placed  in  position  the  ends  of  the  tits  are 
riveted  over  to  hold  it  securely  in  place. 

Machine  screws  are  sometimes  used  in  place  of  tits,  but 
they  suffer  from  the  disadvantage  of  jarring  loose. 

A  satisfactory  way  of  holding  a  key  in  a  hub  is  shown  in 
Fig.  104. 

Where  the  end  of  a  stud  is  to  receive  change-gears  a  con- 
venient form  of  key  is  the  dovetail  shown  in  Fig.  105  in  cross- 


FIG.  103. 


FIG.  105. 


section.     The  dovetailed  key-seat  is  generally  cut  with  a  mill- 
ing-cutter, and  is  made  a  tight  fit  for  the  key.     After  the  latter 
is  in  place  the  shaft  is  calked  against  it  at  A-A. 
For  feathers,  Richards  gives: 


•w  = 

t  = 


1 


TABLE  XI. 

2}  2 

A  f 

A  \ 


107.  Round  Taper  Keys.  —  For  keying  hand-wheels  and 
other  parts  that  are  not  subjected  to  very  great  stress,  a  cheap 
and  satisfactory  method  is  to  use  a  round  taper  key  driven 
into  a  hole  drilled  in  the  joint,  as  in  Fig.  io6A .  If  the  two  parts 
are  of  different  material,  one  much  harder  than  the  other, 


176 


MACHINE  DESIGN. 


this  method  should  not  be  used,  as  it  is  almost  impossible  in 
such  case  to  make  the  drill  follow  the  joint.  For  these  keys 
it  is  customary  to  use  Morse  standard  tapers,  as  reamers  are 
then  readily  obtainable. 

108.  A  Cotter  is  a  key  that  is  used  to  attach  parts  that  are 
subjected  to  a  force  of  tension  or  compression  tending  to  sepa- 


FIG.  106. 

rate  them.  Thus  piston-rods  are  often  connected  to  both  pis- 
ton and  cross-head  in  this  way.  Also  the  sections  of  long 
pump-rods,  etc.  Fig.  106,  B,  shows  a  taper  pin  cotter. 

Fig.   107  shows  machine  parts  held  against  tension  by  cot- 
ters.    It  is  seen  that  the  joint  may  yield  by  shearing  the  cot- 


ter at  AB  and  CD;  by  shearing  at  CPQ  and  ARS;  by  shearing 
at  MO  and  LN;  by  crushing  at  BD,  or  AR  and  CP;  or  by 
tensile  rupture  of  the  rod  on  the  net  section  at  LM.  All  of  these 
sections  should  be  sufficiently  large  to  resist  the  maximum  stress 
safely.  The  usual  difficulty  is  to  get  sufficient  tensile  strength  at 
LM;  but  this  may  be  accomplished  by  making  the  rod  larger 


MEANS  FOR   PRESENTING  RELATIVE  ROTATION. 


'77 


or  the  cotter  thinner  and  higher.  It  is  found  that  taper  sur- 
faces if  they  be  smooth  and  somewhat  oily  will  just  cease  to 
stick  together  when  the  taper  equals  1.5  inches  per  foot.  The 
taper  of  the  rod  in  Fig.  107  should  be  about  this  value  in  order 
that  it  may  be  removed  conveniently  when  necessary 

From  consideration  of  the  laws  of  friction  it  is  obvious  that 
where  a  taper  cotter  is  used,  either  alone  as  in  Fig.  108  or  in 
connection  with  a  gib  as  in  Fig.  109,  the  angle  of  taper  a  must 
not  exceed  the  friction  angle  <£.  That  is,  if  the  coefficient  of 
friction  be  //,  then  /*  =  tan  (f>  and  tan  a  must  be  less  than  tan  <f> 
or  //.  Since,  for  oily  metallic  surfaces,  p  may  have  a  value 


^  %%ft$$$$$s$$$$$%%  ~\ 
• — * — I 


FIG.  1 08. 


FIG.  109. 


as  low  as  0.08,  it  follows  that  a  must  not  exceed  4^°.  If  both 
surfaces  of  the  cotter  slope  with  reference  to  the  line  of  action 
of  the  force,  the  total  angle  of  the  sloping  sides  must  not 
exceed  9°. 

109.  Set-screws  (see  §  86,  p.  140)  are  frequently  used  to  pre- 
vent relative  motion.  They  are  inadvisable  for  heavy  duty 
and  if  used  on  rotating  members  never  should  have  projecting 
heads.  Experiments  made  by  Professor  Lanza  *  with  f -inch 
wrought-rron  set-screws,  ten  threads  to  the  inch  and  tightened 
with  a  pull  of  75  Ibs.  at  the  end  of  a  lo-inch  wrench,  gave  the 
following  results : 


*  Trans.  A.  S.  M.  E.,  Vol.  X. 


I78  MACHINE  DESIGN. 

TABLE  XII. 

ir-    •.     t  T7_  j  Holding  Power  at 

Kind  of  End.  Surface  of  Shaft. 

Ends  perfectly  flat,  -£§  inch  diameter.  .  .  Average  2064  Ibs. 

Rounded  ends,  radius  \  inch  • "         2912     " 

Rounded  ends,  radius  \  inch "         2573     " 

Cup-shaped  and  case  hardened "         2470    " 

1 10.  Shrink  and  Force  Fits.  —  Relative  rotation  between 
machine  parts  is  also  prevented  sometimes  by  means  of  shrink 
and  force  fits.  In  the  former  the  shaft  is  made  larger  than 
the  hole  in  the  part  to  be  held  upon  it,  and  the  metal  surround- 
ing the  hole  is  heated,  usually  to  low  redness.  Because  of 
the  expansion  it  may  be  put  on  the  shaft,  and  on  cooling  it 
shrinks  and  "grips  '.'  the  shaft.  A  key  is  sometimes  used  in 
addition  to  this.  The  coefficient  of  linear  expansion  for  each 
degree  Fahrenheit  is  0.0000065  for  wrought  iron  and  steel  and 
0.0000062  for  cast  iron.  Low  redness  corresponds  to  about 
600°  F.  and  therefore  causes  an  expansion  of  the  bore  of  about 
0.004  mcn  per  inch  of  diameter.  Were  a  hub  expanded  this 
amount  and  placed  on  an  incompressible  shaft  of  identical  diameter 
so  that  the  fibers  immediately  surrounding  the  bore  could  not 
shrink  as  the  hub  cooled,  the  unit  strain  in  these  fibers  when 
cool  would  be  the  entire  .004  inch  per  inch;  and,  since  unit  stress 
equals  unit  strain  multiplied  by  the  coefficient  of  elasticity,  the 
unit  stress  in  the  bore  fibers  would  equal . 004  X  15,000,000  =  60,000 
Ibs.  per  square  inch  for  cast  iron;  or  .004X30,000,000=120,000 
Ibs.  per  square  inch  for  steel.  This  indicates  that  the  hub  would 
rupture,  beginning  at  the  bore  fibers,  where  the  unit  stress  is 
greatest,  and  extending  outward.  There  is,  however,  no  such 
thing  as  an  incompressible  shaft  material.  Actually  the  bore 
fibers  will  not  be  extended  the  full  amount  of  the  difference  be- 
tween the  original  circumferences  at  the  same  temperature  of 
the  shaft  and  the  hub-bore.  The  relative  compression  and  ex- 
tension of  the  two  members  depends  upon  several  factors  such 
as:  the  value  of  Poisson's  ratio  and  the  coefficient  of  elasticity 


MEANS  FOR   PREVENTING  RELATIVE  ROTATION.          17$ 

(Young's  modulus)  of  the  material  of  each;  whether  the  shaft 
is  solid  or  hollow,  and,  in  the  latter  case,  upon  the  relation  of 
inner  to  outer  diameter;  and  the  relation  of  diameter  of  bore  of 
hub  to  its  outside  diameter. 

It  is  obvious  that  in  the  case  of  shrink  fits  the  bore  expansion 
must  be  sufficient  to  allow  it  to  pass  freely  over  the  originally 
large  shaft  to  its  final  position.  This  in  itself  precludes  the 
possibility  of  using  so  large  an  original  difference  as  .004  inch 
per  inch  of  diameter,  even  if  the  certainty  of  rupture  with  such 
an  allowance  did  not  exist.  An  allowance  of  .001  inch  per  inch 
of  diameter,  or,  as  a  maximum  one  of  .0012  inch  per  inch,  will 
prove  ample  and  will  correspond  to  as  great  fiber  stresses,  with 
ordinary  hub  proportions,  as  it  is  safe  to  use  with  either  cast 
iron  or  steel. 

Force  fits  are  made  in  the  same  way  except  that  they  are 
put  together  cold,  either  by  driving  together  with  a  heavy  sledge 
or  by  forcing  together  by  hydraulic  pressure. 

In  general,  pressure  fits  are  not  employed  on  diameters 
exceeding  10  ins.,  shrinkage  fits  being  used  for  large  work. 

The  alignment  should  be  absolutely  accurate  in  starting. 
To  secure  this  some  engineers  resort  to  the  use  of  two  diameters 
— each  half  the  length  of  the  fit — differing  by  but  a  few  thou- 
sandths of  an  inch,  or  taper  the  first  half-length  an  amount  equal 
to  the  allowance.  The  surfaces  should  be  as  smooth  as  possible 
and  well  lubricated.  Linseed  oil  is  recommended. 

Experience  shows  that,  for  same  allowances,  shrink  fits  hold 
more  firmly  than  pressure  fits.* 

in.  Stress  in  Hub. — In  regard  to  the  tension  in  the  hub 
due  to  shrinkage  or  forced  fits,  completely  satisfactory  data 
are  lacking.  A  close  approximation  to  the  probable  tension 
in  the  inner  layer  of  the  hub — the  one  next  the  shaft — may  be 

*  For  valuable  data  on  fits  ancl  fittings,  see  Am.  Mach.,  Mar.  7,  1907;  Trans, 
A.  S.  M.  E.,  Vols.  24,  34  and  35;  Halsey's  "Handbook  for  Machine  Designers." 


l8o  MACHINE    DESIGN. 

made  by  considering  the  hub  as  a  thick  cylinder  under  internal 
pressure.  Using  Lame's  analysis,  Professor  Morley*  develops 
the  following  formulae : 

£s 


m~I  ,  I 

m       mi 


d     EI 


-i        i 


mE 


P2=-    -, .     (3) 

2000 

P-pP* (4) 

/  —  unit  tensile  fiber  stress  at  bore,  pounds  per  square  inch; 
/>2  =  unit  radial  pressure  on  bore  surface,  pounds  per  square  inch; 

A=  total  fit  allowance;    excess  of  original  shaft  diameter  over 
that  of  original  bore,  inches; 

d= original  bore  diameter,  inches; 
/= original  length  at  hub,  inches; 

fi=  outer  radius  of  hub,  inches; 

f2  =  inner  radius  of  hub,  inches ; 

E  =  Young's  modulus,  shaft  material; 
EI  =  Young's  modulus,  hub  material; 

—  =  Poisson's  ratio,  shaft  material; 
m 

—  =  Poisson's  ratio,  hub  material; 
mi 

*  Engineering,  Aug.  n,  1911. 


MEANS  FOR   PREVENTING  RELATIVE  ROTATION.  iSi 

j«  =  coefficient  of  friction  between  shaft  and  hub  materials; 
PZ  =  total  normal  pressure  between  shaft  and  hub-bore  surface, 

tons. 
P  =  total  pressure  required  to  force  shaft  into  hub,  tons. 

Problem. — A  steel  shaft  5  ins.  in  diameter  is  to  be  forced 
into  a  cast-iron  hub  10  ins.  in  outer  diameter  and  8  ins.  long. 
Allowing  .001  inch  per  inch  excess  diameter  of  shaft  over  bore 
for  the  force  fit,  what  will  be  the  unit  tensile  stress  at  the  bore 
of  the  hub,  and  what  will  be  the  necessary  forcing  pressure  ? 


—  =.ooi; 
a 

£=29,000,000  for  steel; 
EI  =  14,500,000  for  cast  iron; 

-=.299  for  steel;  ^  =  3.35; 
m 

—  =  .274  for  cast  iron;  7^  =  3.65; 


. 


:  =  o.i25J 

.001X29,000,000 


i     29,000,000X25-6.25     29,000,000  1/ 


3-35     3-65 
10,545  Ibs.  per  sq.in.  ; 

I0'545 


.001- 


14,500,000 


3.35  X  29,000,000     3.65  X  14,500,000 
=  6350  Ibs.  per  square  inch; 


2000 

P=  o.i  25X399  =  50  tons,  nearly. 


182 


MACHINE  DESIGN. 


i.o 


I  For  usual  cases'  with  a  comparatively  eniall  hole  in  outer  ring, 
I  infafiLin_a  wh_ee_l_of_any  shape  with  a  solid  shaft,  or  shaft  with, 
n  even  a  considerable  hole,  the  bore  of  the  outer  ring  expands 
-  -  about  0.6  of  the  force  fit.- 


OJ 


0.2- 


0.1          0.2          0.3          0.4          0.5          0.6          0.7          0.8 
Ratio  of  Bore  Diam.  to  Outer  Diam.  of  Outer  Ring 


0.9 


FIG.  109^4. — RATIO  OF  EXPANSION  OF  BORE  TO  FORCE  FIT. 
Figures  on  Curves  Denote  Ratio  of  Diameter  of  Hole  in  Inner  Ring  (i.e.,  Shaft) 
to  Diameter  of  Bore  of  Outer  Ring  (i.e.,  Hub). 


14,000 


J>  a,  12,000 


0.1 


0.2          0.3          0.4          0.5          0.6          0.7          0.8          0.9 
Ratio  of  Bore  Diam.  to  Outer  Diam.  of  Outer  Ring 


IJf 


FIG.  logB. — RADIAL  STRESS  AT  BORE.     FORCE  FIT  OF  o.ooi  IN.  PER  IN.  OF  BORE. 
Stress  is  directly  proportional  to  Force  Fit  per  Inch    of    Bore.     Figures  on 
Curves  Denote  Ratio  of  Diameter  of  Hole  in  Inner  Ring  to  Diameter  of  Bore 
of  Outer  Ring. 


MEANS  FOR  PREVENTING  RELATIVE  ROTATION. 


18? 


112.  Hub  Stresses  by  Guest's  Law.— Mr.  Sanford  A.  Moss  * 
has  developed  formulas  based  upon  Guest's  maximum  shear 
law  and  the  use  of  the  same  material  (steel)  in  outer  and  inner 
rings,  i.e.,  hub  and  shaft.  He  assumes  also  that  the  shaft  itself 
may  be  hollow  as  well  as  solid.  This  theory  gives  greater  values 
to  the  effective  stresses  than  does  the  Lame  theory.  Figs.  109,4. 
to  icgC  show  the  results  in  chart  form. 


u.i 


0.2          0.3          0.4          0.5          0.6          0.7          0.8          0.9 
Hatio  of  Bore  Diam.  to  Outer  Diam.  of  Outer  Ring 


FIG.   iooC.— EFFECTIVE  TENSILE  STRESS  AT  BORE  OF  OUTER  RING.    FORCE 

FlT   OF    O.OOI    IN.    PER   IN.    OF   BORE. 

Stress  is  Directly  Proportional  to  Force  Fit  per  Inch  of  Bore.  Figures  on 
Curve  Denote  Ratio  of  Inner  Diameter  of  Inner  Ring  to  Bore  Diameter. 

To  compute  the  forcing  pressure  necessary  with  these  effective 
radial  stresses  a  value  of  ^=.038,  obtained  from  actual  tests, 
is  recommended.  This  low  value  of  fi  indicates  that  the  value 
of  the  radial  stress  computed  by  this  theory  is  too  high.  The 
same  is  probably  true  of  the  effective  tensile  stress. 

*  Trans. 'A,  S.  M.  E.,  Vol.  34. 


1 84  MACHINE  DESIGN. 

Fig.  109^  can  be  used  to  determine  the  approximate  unit 
strain  for  any  given  materials  and  proportions  of  hub  and  shaft. 
From  this  strain  the  corresponding  stress  can  be  computed 
directly.  In  the  problem  considered  in  the  preceding  section 
the  ratio  of  bore  diameter  to  outer  diameter  of  hub  =  0.5.  The 
shaft  being  solid,  the  chart  shows  a  ratio  of  expansion  of  bore 
to  force  fit  allowance  of  .735,  found  by  following  up  the  ordi- 
nate  at  0.5  until  it  intersects  the  curve  for  solid  shafts.  For  a 
unit  fit  allowance  of  .001  inch,  then,  the  bore  fibers  in  this  case 
are  strained  .000735  mc^  Per  inch.  With  a  coefficient  of  elas- 
ticity of  14,500,000,  this  corresponds  to  a  unit  tensile  stress  of 
10,650  Ibs.  This  checks  with  the  result  obtained  from  Morley's 
formula.  Had  the  shaft  been  bored  out  with  an  internal  diameter 
of  .5  its  outer  diameter,  the  hub  remaining  the  same  as  before, 
the  ratio  of  bore  expansion  to  unit  force  fit  allowance  would  have 
been  but  .59.  The  unit  strain  in  this  case  would  have  been 
.00059  inch  per  inch,  and  the  corresponding  stress, 

14,500,000 X. 00059  =  85 50  Ibs.  per  square  inch. 


CHAPTER  X. 


SLIDING   SURFACES. 

113.  General  Discussion. — So  much  of  the  accuracy  of 
motion  of  machines  depends  on  the  sliding  surfaces  that  their 
design  deserves  the  most  careful  attention.  The  perfection  of 
the  cross-sectional  outline  of  the  cylindrical  or  conical  forms 
produced  in  the  lathe  depends  on  the  perfection  of  form  of  the 
spindle.  But  the  perfection  of  the  outlines  of  a  section  through 
the  axis  depends  on  the  accuracy  of  the  sliding  surfaces.  All 
of  the  surfaces  produced  by  planers,  and  most  of  those  pro- 
duced by  milling-machines,  are  dependent  for  accuracy  on  the 
sliding  surfaces  in  the  machine. 


W/7///////y//////^^^ 
FIG.  no. 


F  Q 

FIG.  in. 


114.  Proportions  Dictated  by  Conditions  of  Wear. — Suppose 
that  the  short  block  A,  Fig.  no,  is  the  slider  of  a  slider-crank 
chain,  and  that  it  slides  on  a  relatively  long  guide  D.  The 
direction  of  rotation  of  the  crank  a  is  as  indicated  by  the  arrow. 
B  and  C  are  the  extreme  positions  of  the  slider.  The  pressure 
between  the  slider  and  the  guide  is  greatest  at  the  mid-position, 
A ;  and  at  the  extreme  positions,  B  and  C,  it  is  onjy  the  pressure 
due  to  the  weight  of  the  slider.  Also  the  velocity  is  a  maximum 
when  the  slider  is  in  its  mid-position,  and  decreases  towards  the 
ends,  becoming  zero  when  the  crank  a  is  on  its  center.  The 
work  of  friction  is  therefore  greatest  at  the  middle,  and  is  very 

185 


1 86 


MACHINE  DESIGN. 


small  near  the  ends.  Therefore  the  wear  would  be  the  greatest 
at  the  middle,  and  the  guide  would  wear  concave.  If  now  the 
accuracy  of  a  machine's  working  depends  on  the  perfection 
of  A's  rectilinear  motion,  that  accuracy  will  be  destroyed  as  the 
guide  D  wears.  Suppose  a  gib,  EFG,  to  be  attached  to  A,  Fig. 
in,  and  to  engage  withZ),  as  shown,  to  prevent  vertical  loose- 
ness between  A  and  D.  If  this  gib  be  taken  up  to  compensate 
for  wear  after  it  has  occurred,  it  will  be  loose  in  the  middle 
position  when  it  is  tight  at  the  ends,  because  of  the  unequal 
wear.  Suppose  that  A  and  D  are  made  of  equal  length,  as  in 

Fig.  112.  Then 
when  A  is  in  the 
mid-position  corre- 

O    [    I -| —    _^     p  A  — '  spending    to  maxi- 

'   mum  pressure   vel- 

FlG.   112.  . 

ocity,  and  wear,  it 

is  in  contact  with  D  throughout  its  entire  surface,  and  the  wear  is 
therefore  the  same  in  all  parts  of  the  surface.  The  slider  retains 


its  accuracy  of  rectilinear  motion  regardless  of  the  amount  of 
wear;  the  gib  may  be  set  up,  and  will  be  equally  tight  in  all 
positions. 

If  A    and  B,  Fig.  113,  are  the  extreme  positions  of  a  slider, 
D  being  the  guide,  a  shoulder  would  be  finally  worn  at  C.     It 


SLIDING  SURFACES.  187 

would  be  better  to  cut  away  the  material  of  the  guide,  as  shown 
by  the  dotted  line,  or,  better  still,  to  cut  a  ratchet  surface  on  the 
guide  as  shown  at  B.  Such  a  surface  aids  lubrication  by  acting 
as  an  oil  reservoir.  Slides  should  always  "  wipe  over  "  the  ends 
of  the  guide  when  it  is  possible.  Sometimes  it  is  necessary 
to  vary  the  length  of  stroke  of  a  slider,  and  also  to  change  its 
position  relatively  to  the  guide.  Examples:  "Cutter-bars" 
of  slotting-  and  shaping-machines.  In  some  of  these  positions 
there  will  be  a  tendency,  therefore,  to  wear  shoulders  in  the 
guide  and  also  in  the  cutter-bar  itself.  This  difficulty  is  over- 
come if  the  slide  and  guide  are  made  of  equal  length,  and  the 
design  is  such  that  when  it  is  necessary  to  change  the  position 
of  the  cutter-bar  that  is  attached  to  the  slide,  the  position  of 
the  guide  may  be  also  changed  so  that  the  relative  position  of 
slide  and  guide  remains  the  same.  The  slider  surface  will  then 
just  completely  cover  the  surface  of  the  guide  in  the  mid-position, 
and  the  slider  will  wipe  over  each  end  of  the  guide  whatever 
the  length  of  the  stroke.* 

In  many  cases  it  is  impossible  to  make  the  slider  and  guide 
of  equal  length.  Thus  a  lathe-carriage  cannot  be  as  long  as  the 
bed,  a  planer-table  cannot  be  as  long  as  the  planer-bed,  nor  a 
planer-saddle  as  long  as  the  cross-head.  When  these  condi- 
tions exist  especial  care  should  be  given  to  the  following: 

I.  The  bearing  surface  should  be  made  so  large  in   pro* 
portion  to  the  pressure  to  be  sustained  that  the  maintenance 
of  lubrication  shall  be  insured  under  all  conditions. 

II.  The  parts  which  carry  the  wearing  surfaces  should  be 
made  so  rigid  that  there  shall  be  no  possibility  of  the  localiza- 
tion of  pressure  from  yielding. 

115.  Form  of  Guides. — As  to  form,  guides  may  be  divided 
into  two  classes:  angular  guides  and  flat  guides.  Fig.  114,  a, 
shows  an  angular  guide,  the  pressure  being  applied  as  shown. 

*  See  further,  Things  That  Are  Usually  Wrong,  by  Prof.  J.  E.  Sweet. 


1 88  MACHINE  DESIGN. 

The  advantage  of  this  form  is,  that  as  the  rubbing  surfaces 
wear,  the  slide  follows  down  and  takes  up  both  the  vertical 
and  lateral  wear.  The  objection  to  this  form  is  that  the  pres- 
sure is  not  applied  at  right  angles  to 
the  wearing  surfaces,  as  it  is  in  the 
flat  guide  shown  in  b.  But  in  b,  a  gib 
must  be  provided  to  take  up  the  lateral 
wear.  The  gib  is  either  a  wedge  or 
a  strip  with  parallel  sides  backed  up  by  screws  and  offers 
danger  of  localized  pressure  or  binding.  Guides  of  these  forms 
are  used  for  planer-tables.  The  weight  of  the  table  itself  holds 
the  surfaces  in  contact,  and  if  the  table  is  light  the  tendency 
of  a  heavy  side  cut  would  be  to  force  the  table  up  one  of  the 
angular  surfaces  away  from  the  other.  If  the  table  is  very  heavy, 
however,  there  is  little  danger  of  this,  and  hence  the  angular 
guides  of  large  planers  are  much  flatter  than  those  of  smaller 
ones.  In  some  cases  one  of  the  guides  of  a  planer-table  is 
angular  and  the  other  is  flat.  The  side  bearings  of  the  flat  guide 
may  then  be  omitted,  as  the  lateral  wear  is  taken  up  by  the 
angular  guide.  This  arrangement  is  undoubtedly  good  if  both 
guides  wear  down  equally  fast. 

As  regards  lathe  beds,  while  English  practice  favors  the  flat 
way,  the  form  of  guide  used  in  America  is  almost  exclusively 
the  inverted  V.  The  angle  for  small  lathes  is  90°  and  increases 
slightly  in  obtuseness  with  the  larger  sizes.  The  main  advantage 
of  the  inverted  V  form  is  that  of  automatic,  certain  adjustment, 
the  adjustment  with  flat  ways  depending  upon  the  use  of  gibs. 
On  the  other  hand,  the  flat  ways  offer,  a  more  distributed  sup- 
port to  the  carriage,  which  is  therefore  less  apt  to  spring  under 
the  stress  of  a  heavy  cut  than  if  supported  solely  at  the  outer  Vs. 
Fig.  114^4  shows  a  modern  compromise  as  adopted  on  a  small  lathe. 
Fig.  115  shows  several  forms  of  sliding  surfaces  such  as  are 
used  for  lathes,  shapers,  milling  machines,  etc. 


SLIDING  SURFACES. 

JL-U4 


189 


1/,'Hold.down  Elate      S£^    tfkjlF- 
parallel  gib 
J$"set  screw' 


and  lock  nut 


FIG. 


FIG.  115, 


19°  MACHINE  DESIGN. 

At  A  is  shown  a  form  employing  a  taper  gib,  usually  made 
of  bronze,  adjusted  by  means  of  a  stud  and  lock  nuts,  d.  The 
angle  0  is  generally  60°,  although  it  is  sometimes  made  as  acute 
as  45°.  The  form  shown  at  B  also  employs  a  taper  gib,  but  it 
is  set  up  by  means  of  a  headless  screw  engaged  in  a  thread  cut 
in  the  slide.  The  threads  are  cut  out  of  the  corresponding 
portion  of  the  gib,  and  the  end  of  the  screw,  pressing  against 
the  gib  at  the  bottom  of  this  recess,  forces  the  gib  in  as  the  screw 
is  turned.  The  adjustments  of  C,  D,  and  E  are  obvious.  These 
heavier  gibs  are  usually  made  of  cast  iron;  they  are  not  tapered 
lengthwise.  C  is  the  most  accessible,  but  least  stable.  F  em- 
ploys a  cast-iron  gib,  tapered  and  adjusted  by  means  of  a  screw 
which  has  a  broad  collar  or  disk.  G  and  H  employ  parallel 
thin  gibs  of  bronze  or  steel,  set  up  by  means  of  adjusting  screws. 
This  form  is  not  so  satisfactory  as  the  wedge  gib,  as  the  bearing  is 
chiefly  under  the  points  of  the  screws,  the  gib  being  thin  and  yield- 
ing, whereas  in  the  wedge  there  is  complete  contact  between  the 
metallic  surfaces.  /  shows  an  arrangement,  employing  no  angular 
surfaces,  such  as  is  used  on  some  shaping  machines  to  constrain  the 
ram.  K  shows  an  arrangement  employed  on  planer  cross-rails. 

The  sliding  surfaces  thus  far  considered  have  to  be  designed 
so  that  there  will  be  no  lost  motion  while  they  are  moving 
because  they  are  required  to  move  while  the  machine  is  in 
operation.  The  gibs  have  to  be  carefully  designed  and  accu- 
rately set  so  that  the  moving  part  shall  be  just  "  tight  and 
loose  ";  i.e.,  so  that  it  shall  be  free  to  move,  without  lost  motion 
to  interfere  with  the  accurate  action  of  the  machine.  There  is, 
however,  another  class  of  sliding  parts,  like  the  sliding-head 
of  a  drill-press,  or  the  tailstock  of  a  lathe,  that  are  never  required 
to  move  while  the  machine  is  in  operation.  It  is  only  required 
that  they  shall  be  capable  of  being  fastened  accurately  in  a 
required  position,  their  movement  being  simply  to  readjust  them 
to  other  conditions  of  work  while  the  machine  is  at  rest.  No 


SLIDING  SURFACES.  191 

gib  is  necessary  and  no  accuracy  of  MOTION  is  required.  It 
is  simply  necessary  to  insure  that  their  position  is  accurate 
when  they  are  clamped  for  the  special  work  to  be  done. 

116.  Lubrication. — The   question  of  strength   rarely  enters 
into  the  determination  of  the  dimensions  of  sliding  surfaces; 
these  are  determined   rather  by  considerations   of  minimizing 
wear  and  maintaining  lubrication.     As  long  as  a  film  of  oil 
separates  the  surfaces,  wear  is  reduced  to  a  minimum.     The 
allowable   pressure   between    the   surfaces   without   destruction 
of  the   film   of  lubricant   varies   with  several   conditions.     To 
make  this  clear,  suppose  a  drop  of  oil  to  be  put  into  the  middle 
of  an  accurately  finished  surface  plate  (i.e.,  as  close  an  approxi- 
mation to  a  plane  surface  as  can  be  produced) ;  suppose  another 
exactly  similar  plate  to  be  placed  upon  it  for  an  instant;    the 
oil-drop  will  be  spread  out  because  of  the  force  due  to  the  weight 
of  the  upper  plate.     Had  the  plate  been  heavier  it  would  have 
been  spread  out  more.     If  the  plate  were  allowed  to  remain 
a  longer  time,  the  oil  would  be  still  further  spread  out,  and  if 
its  weight  and  the  time  were  sufficient,  the  oil  would  finally  be 
squeezed  entirely  out  from  between  the  plates,  and  the  metal 
surfaces  would  come  into  contact.     The  squeezing  out  of  the  oil 
is,  therefore,  a  function  of  the  time  as  well  as  of  pressure. 

If  the  surfaces  under  pressure  move  over  each  other,  the 
removal  of  the  oil  is  facilitated.  The  greater  the  velocity  of 
movement  the  more  rapidly  will  the  oil  be  removed,  and  there- 
fore the  squeezing  out  of  the  oil  is  also  a  function  of  the  velocity 
of  the  rubbing  surfaces. 

117.  Allowable  Bearing  Pressure. — Flat  surfaces  in  machines 
are  particularly  difficult  to  make  perfectly  true  in  the  first  place, 
and  to  keep  true  in  the  course  of  operation  of  the  machines. 
If  they  are  distorted  ever  so  slightly  the  pressure  between  the  sur- 
faces becomes  concentrated  at  one  small  area,  and  the  actual  pres- 
sure per  square  inch  is  vastly  in  excess  of  the  nominal  pressure. 


1 92  MACHINE   DESIGN. 

In  consequence  of  this  and  the  differences  in  original  truth 
and  finish  of  the  surfaces,  there  is  no  matter  in  machine  design 
in  which  practice  varies  more  than  in  the  nominal  pressure 
allowed  per  square  inch  of  bearing  area  of  flat  sliding  surfaces. 

Unwin  *  gives  the  following: 

TABLE  XIV. 

Cast  iron  on  babbitt  metal 200  to  300  Ibs.  per  square  inch 

Cast  iron  on  cast  iron,  slow 60  to  100  Ibs.  per  square  inch 

Cast  iron  on  cast  iron,  fast 40  to    60  Ibs.  per  square  inch 

Professor  Barr  f  found  American  practice  to  vary  as  follows : 

Cross-head  shoes  of  high-speed  engines: 

Minimum  pressure  per  square  inch 10.5  Ibs. 

Maximum  pressure  per  square  inch 38        " 

Mean  pressure  per  square  inch 27        " 

Cross-head  shoes  of  low-speed  engines: 

Minimum  pressure  per  square  inch 29  Ibs. 

Maximum  pressure  per  square  inch 58    ' ' 

Mean  pressure  per  square  inch 40    " 

In  all  cases  the  mean  sliding  velocity  was  probably  in  the 
neighborhood  of  600  feet  per  minute  with  a  maximum  velocity 
at  the  middle  of  the  stroke  of  about  950  feet  per  minute.  In 
"  low- speed  "  engines  the  maximum  velocity  is  only  reached 
about  one  third  as  many  times  per  minute  as  in  "  high-speed  " 
engines,  although  they  may  have  the  same  mean  velocity,  and 
it  is  therefore  proper  to  allow  a  higher  unit  value  of  pressure 
for  the  former  than  for  the  latter.  For  well-made  surfaces  the 
maximum  values  given  by  Professor  Barr  may  be  safely  used. 

For  lower  mean  speeds  than  600  feet  per  minute  they  may 
be  increased,  and  for  higher  speeds  decreased,  according  to  some 
law  such  as  ^7  =  36,000, 

in  which  formula  p  =  pressure  per  square  inch,  and  V  =  velocity 
of  rubbing  in  feet  per  minute. 

*  "Machine  Design,"  1909,  Vol.  I,  p.  252. 
t  Trans.  A.  S.  M.  E.,  Vol.  XVIII,  p.  753. 


SLIDING   SURFACES 


193 


118.  Maintenance  of  Lubrication.— Regarding  the  materials 
to  be  used,  brass,  bronze,  or  babbitt  metal  will  run  well  with 
iron  or  steel.  Cast  iron  on  cast  iron  is  frequently  used,  par- 
ticularly in  machine  tool  construction. 

To  maintain  lubrication  a  constant  flow  of  oil  from  a  cup  is 
desirable.  The  moving  surface  should,  if  possible,  have  chan- 
nels cut  in  its  face  to  conduct  the  oil  from  the  central  oil-hole 
to  all  parts  of  the  surface,  as  shown  in  Fig.  116.  The  layout 
of  the  oil-grooves  should  be  such  as  will  take  advantage  of  the 


FIG.  116. 


1 

I 

:'- 

1 

1 

1 

FIG.  117. 


FIG.  118. 


motion  of  the  part  to  draw  the  oil  along  them  and  distribute 
it  over  the  entire  bearing  area.  The  oil  should  be  forced  in  where 
the  pressures  are  heavy. 

Oil-pads  may  be  used  as  shown  in  Fig.  117.  The  shaded 
areas  represent  porous  pads  whose  lower  surfaces  just  touch  the 
surface  to  be  lubricated,  and  which  are  kept  soaked  with  oil. 

For  oiling  the  ways  of  planer-tables  it  is  customary  to  use 
rollers  placed  in  oil- filled  pockets  in  the  guides.  The  top  of 
the  roller  is  held  against  the  surface  of  the  way  by  means  of 
springs.  (See  Fig.  118.) 

It  is  extremely  difficult  to  maintain  continuous  film  lubri- 
cation with  reciprocating  surfaces  except  with  forced  lubrication. 
In  some  cases  it  may  be  possible  to  employ  the  principle  of  the 
Kingsbury  thrust  bearing.  (See  p.  246.) 


CHAPTER  XL 


AXLES,   SHAFTS,  AND   SPINDLES. 

119.  By  Axles,  Shafts,  or  Spindles  we  denote  those  rotating 
or  oscillating  members  of  machines  whose  motion  is  constrained 
by  turning  pairs.      Axle  is  the  name   given  to  such  a  member 
when  it  is  subjected  to  a  load  which  produces  a  bending  moment, 
and  the  only  torsional  stress  is  that  due  to  friction. 

When  rotating  members  are  subjected  chiefly  to  torsional 
stress,  or  combined  torsion  and  bending,  they  are  called  shafts 
or  spindles.  The  former  term  is  used  where  the  part  has  as 
its  function  the  transmission  of  energy  of  rotation  from  one 
point  to  another.  Examples  are  line-shafts  and  crank-shafts. 

The  term  spindle,  on  the  other  hand,  is  restricted  to  those 
rotating  members  which  are  directly  connected  with  the  tool  or 
work  and  give  it  an  accurate  rotative  motion.  They  generally 
form  the  main  axis  of  the  machine.  Examples  are  the  lathe-  and 
drill-spindles. 

120.  Axle  Design. — The  question  of  axle  design  will  be  taken 
up  first,  and  the  torsional  moment  due  to  friction  will  be  neglected. 


FIG.  119. 

A  typical  case  is  shown  in  Fig.  119.  Here  the  two  ends  are 
purposely  not  symmetrical.  Given  the  loads  P  and  Q,  solution 
is  first  made  for  the  reactions  R  and  S  by  the  ordinary  methods 

of  mechanics,  grauhical  or  analytical. 

194 


AXLES,  SHAFTS,  AND  SPINDLES. 


195 


The  graphical  method  is  best,  since  it  gives  the  moments  at 
all  sections.  Lay  off  the  line  M-N,  Fig.  120,  whose  length 
equals  /'  +  /  +  /",  the  distance  between  the  points  of  applications 
of  R  and  5.  Denote  the  points  of  application  of  R,  P,  Q,  and 
S  by  a,  b,  c,  and  d,  respectively.  At  b  erect  a  perpendicular  and 


R 

t-f- 


CO 

4 


FIG.  120. 


lay  off  on  it  a  vector  representing  the 
value  of  P  in  pounds.  At  c  erect  a 
perpendicular  and  lay  off  on  it  a  vector  representing  Q  on  the 
same  scale.  At  d  drop  a  perpendicular  and  lay  off  de  equal 
to  vector  Q,  and  ef  equal  to  vector  P.  Select  any  point  O 
as  pole,  and  draw  Od,  Oe,  and  Of.  Denote  by  g  the  point 
where  Od  intersects  a  perpendicular  dropped  from  c,  and  draw 
from  g  a  parallel  to  Oe  until  it  intersects  a  perpendicular  dropped 
from  b  at  h.  From  h  draw  a  parallel  to  Of  until  it  intersects  a 
perpendicular  dropped  from  a  at  j.  Draw  /d,  and  parallel  to  jd 
draw  a  line  through  O.  This  line  cuts  the  perpendicular  dropped 
from  d  at  the  point  k.  Then  vector  fk  =  R,  and  kd  =  S,  on 
the  same  scale  as  was  originally  used  for  P  and  Q.  Values 
of  R  and  S  in  pounds  are  therefore  determined.  The  shaded 
area  dghjd  is  the  moment  diagram.  The  vertical  ordinates 
included  between  its  bounding  lines  are  proportional  to  the 
moments  at  the  corresponding  points. 

The  scale  of  the  moment  diagram  can  be  readily  determined 
by  solving  for  the  actual  moment  for  one  section.  Select  the  sec- 
tion at  b.  The  moment  Mb  is  represented  by  mh  and  has  a 
value  =Rl't  R  being  expressed  in  pounds  and  /'  in  inches;  the 


196  MACHINE  DESIGN. 

value  of  the  moment  in  inch-pounds  can  be  determined  at  any 
point,  since  the  scale  used  is  mh  inches  equals  Rlf  inch-pounds. 
For  a  circular  section  we  have  the  elastic  moment 


c         4 

Mb  is  the  bending  moment  in  inch-pounds; 
ft  is  the  unit  stress  in  outer  fiber  in  pounds  per  square  inch; 
/  is  the  plane  moment  of  inertia  of  the  section  in  biquad- 

ratic inches; 

c  is  the  distance  from  neutral  axis  to  outermost  fiber  in 
inches. 

Equating  this  to  the  various  selected  values  of  Mb  and  solv- 
ing for  r  gives  the  radius  of  the  axle  at  any  point. 

In  designing  axles,  great  care  must  be  taken  that  all  forces 
acting  are  being  considered,  and  that  the  maximum  value  of 
each  is  selected.* 

Thus  it  has  been  found  that  the  force  due  to  vertical  oscilla- 
tion caused  by  jar  in  running  is  about  40  per  cent  of  the  static 
load  for  car-axles.  The  axles  would  therefore  have  to  be  de- 
signed for  a  load  1.4  times  the  static  load.  In  addition  to  this 
there  is  in  car-axles  a  bending  moment  due  to  curves,  switches, 
and  wind-pressures.  This  may  amount  as  a  maximum  to  the 
equivalent  of  a  horizontal  force  H,  equal  to  40  per  cent  of  the 
static  load,  applied  at  a  height  of  6  feet  from  the  rail. 

When  such  a  careful  analysis  of  the  forces  has  been  made, 
jt  may  be  taken  for  good  material,  equal  to  one  fourth  of  the 
ultimate  strength,  this  being  a  safe  value  for  cases  like  this, 
where  the  fibers  are  subjected  to  alternate  tension  and  com- 
pression as  determined  by  Wohler  and  others. 

Had  the  static  load  alone  been  considered  in  the  calculations, 

*  See  further  Proc.  Master  Car-Builders'  Assn.,  1896;  Report  of  Committee 
on  Axles,  etc.  Also  Strength  of  Railway-Car  Axles,  Trans.  A.  S.  M.  E.,  1895. 
Reuleaux,  "The  Constructor,"  trans,  by  H.  H.  Supplee,  Philadelphia,  1893; 
Railway  Machinery,  Mar.  1907. 


AXLES,  SHAF1S,  AND  SPINDLES.  IQ7 

ft  should  not  have  been  taken  greater  than  one  tenth  of  the 
ultimate  strength. 

121.  Shafting  Subject  to  Simple  Torsion.  —  If  a  short  shaft 
is  subjected  to  simple  torsion,  its  diameter  may  be  deter- 
mined very  readily  by  the  simple  formula  for  torsional 
moment, 


Here  Mt  =  Pa  is  the  torsional  moment,  P  being  the  force  tending 
to  twist  the  piece  in  pounds  and  a  being  the  lever-arm  of  P 
about  the  axis  of  the  piece  in  inches. 

/  is  the  polar  moment  of  inertia  of  the  cross-section  of  the 

member  in  biquadratic  inches; 
c  is  the  distance  from  the  neutral  axis  to  the  outermost  fiber 

in  inches; 

ja  is  the  allowable  unit  stress  in  pounds  per  square  inch. 
For  a  solid  circular  section 


and 


which  can  be  solved  readily  for  r  ,  the  radius  of  the  shaft. 
For  a  hollow  circular  (i.e.,  ring-shaped)  section, 


C  2Ti 

i  ,      f&  (rl4  " 

and  M  t  = 


Here  TI  is  the  radius  of  the  outside  of  the  shaft  and  r2  is  the 
radius  of  the  bore,  both  in  inches. 

The  way  to  solve  this  is  to  take  r2  as  a  decimal  part  of  r\. 
Thus,  let  r2  =  bri.     It  then  becomes  an  easy  matter  to  solve  for  rL. 


198 


MACHINE  DESIGN. 


122.  Shafting  Subject  to  Combined  Torsion  and  Bending. — 

In  most  cases  shafts  are  subjected  to  combined  torsion  and 
bending.  Consider  the  crank-shaft  shown  in  Fig.  121  in  side 
and  end  view. 


FIG.  121. 


B  is  the  center  of  the  bearing,  C  is  the  center  of  the  crank- 
pin.  At  B  we  have  the  shaft  subjected  to  a  bending  moment, 
Mb  =  Pl,  and  also  to  a  twisting  moment,  Mt=Pa. 

Let  Meb  represent  the  bending  moment  which  would  produce 
the  same  stress  in  the  outer  fiber  as  Mb  and  Mt  combined.  It 
will  be  called  the  equivalent  bending  moment.  Then  it  has 
been  found  *  that 


(I) 


Also, 


(2) 


fnr* 


For  a  circular  section  (2)  becomes  Af«/,= ;  and  substitution 

in  (i)  gives 


,  .     .     .     (3) 


*  See  Bach,  "Elasticitat  und  Festigkeit.'  For  simplicity's  sake,  the  coefficient 
«o  has  been  dropped,  as  it  modifies  the  result  very  slightly  for  wrought  iron  and 
steel.  This  is  based  upon  St.  Venant's  maximum  strain  theory.  Guest  (Phil. 
Mag.,  July,  1900,  p.  132)  advances  the  formula  Meb=^Mb2+Mt2  as  based  upon 
'the  maximum  shear  theory  and  experiments  on  combined  stresses,  but  states 
(p.  78)  that  experimental  results  cannot  be  held  to  disprove  the  maximum  strain 
theory  as  they  do  (p.  77)  the  maximum  stress,  or  Rankine,  theory. 


AXLES,  SHAFTS,  AND  SPINDLES.  199 

which  can  readily  be  solved  for  r,  f  being  given  a  value  equal 
to  the  maximum  allowable  unit  tensile  stress  for  the  material 
and  conditions.* 

For  a  hollow  circular  section 


4/1 

To  solve  this  express  r2  as  a  decimal  part  of  r\.     Substitute  and 
solve  for  r^. 

If  there  are  several  forces  acting,  as  there  are  apt  to  be,  the 
method  is  as  follows:  First,  find  Mb  due  to  all  the  bending 
forces  combined.  Second,  find  Mt  due  to  all  the  twisting  forces 
combined.  Third,  use  these  values  of  Mb  and  Mt  in  equations 
(i),  (3),  and  (4).  Among  the  forces  acting  we  must  not  fail 
to  include  the  weight  of  the  shaft  and  attached  parts. 

123.  Comparison  of  Solid  and  Hollow  Shafts.  —  It  is  evident 
from  (3)  and  (4)  that  the  dimensions  of  a  solid  shaft  and  a 
hollow  shaft  of  equal  strength  will  have  the  relationship 


If  r2=o.6ri,  we  have 


*  Merriman,  "Mechanics,"  nth  ed.,  p.  267,  develops  formulae  on  the  maxi- 
mum shear  theory  which  reduce  to 


and 


in  which  f's  is  the  unit  resultant  shear  on  diagonal  surface  due  to  combination  of 
unit  flexural  tension  and  unit  torsional  shear,  and  /'  is  the  unit  resultant  tensile 
stress  normal  to  the  diagonal  surface.  According  to  Guest's  results,  /'«  must  have 
a  value  less  than  one-half  of  the  unit  tensile  strength  at  "yield  point"  for  ductile 
materials. 


200  MACHINE  DESIGN. 

Hence  a  hollow  shaft  whose  internal  bore  is  0.6  of  its  ex- 
ternal diameter,  in  order  to  have  the  same  strength  as  a  solid 
shaft  must  have  its  external  diameter  1.047  times  the  diameter 
of  the  solid  shaft.  The  weight  of  the  hollow  shaft  will  be  70 
per  cent  of  that  of  the  solid  shaft.  It  is  obvious  that  a  con- 
siderable saving  in  weight  may  be  effected  without  appreciable 
increase  in  size  if  the  hollow  section  is  adopted.  By  using  nickel 
steel  in  connection  with  the  hollow  section  we  can  get  com- 
bined maximum  strength  and  lightness.* 

124.  Angular  Distortion.—  The  angle  by  which  a  shaft  sub- 
jected to  torsion  is  twisted  is  often  an  important  matter.  Let 
this  angle  be  represented  by  fl.  Then 

MthSo 


#  =  angle  of  torsion  in  degrees; 
M  t=  twisting  moment  in  inch-pounds, 
/=  length  of  shaft  in  inches; 
7  =  polar  moment  of  inertia  in  biquadratic  inches; 

modulus  of  elasticity 
G  =  modulus  of  torsion  =  —  —  ^-  . 

2.6 

125.  Combined  Pull  or  Thrust  and  Torsion.  —  When  a  shaft 
is  subjected  to  combined  tension  and  torsion,  or  compression 
and  torsion,  the  following  formulae  have  been  developed  by 
Prof.  Merriman  :  f 

=  2Pd  +  V 


P  =  total  axial  load,  tension  or  compression,  pounds; 
/'  =  unit    normal    resultant    tensile    or    compressive    stress, 
pounds  per  square  inch; 

*See  "Nickel  Steel,"  a  paper  by  D.  H.  Browne  in  Vol.  29  of  the  Trans. 
A.  I.  M.  E. 

f  Mechanics,  Wiley  &  Sons,  1914,  p.  268. 


AXLES,  SHAFTS,  AND  SPINDLES.  201 

//  =  unit  diagonal  resultant  shear  stress,  pounds  per  square  inch; 

Mt  =  twisting  moment,  inch-pounds; 
d=  diameter  of  shaft,  inches. 

In  compression  cases  these  formulae  are  applicable  strictly 
only  to  vertical  shafts  with  supports  sufficiently  close  together  to 
make  it  legitimate  to  consider  them  as  short  columns. 

The  value  of  d  is  found  by  substituting  trial  values  in  the 
equations  until  one  is  determined  which  satisfies  the  identity. 

When  the  distance  between  bearings  is  such  that  the  shaft, 
subjected  to  combined  thrust  and  torsion,  must  be  considered 
as  a  long  column,  the  following  formula  has  been  developed 
by  Prof.  A.  G.  Greenhill:* 


__ 


El    4£2/2 

/=the  length  of  shaft  between  bearings  in  inches; 
P=the  end  thrust  in  pounds; 
E= modulus  of  elasticity; 

/=  plane  moment  of  inertia  of  the  section  in  biquadratic  inches; 
Mt=  twisting  moment  in  inch-pounds. 

This  formula,  also,  is  applicable  strictly  only  to  vertical  shafts, 
as  it  ignores  the  important  item  of  bending  due  to  the  weight 
of  the  shaft  and  attached  parts. 

126.  Line-shafts. — Line-shafts  are  long  shafts  used  to  trans- 
mit power.  They  are  made  of  lengths  coupled  together  and 
supported  by  bearings  at  suitable  intervals.  Pulleys  or  gears 
are  keyed  to  them,  and  should  always  be  placed  as  close  to  the 
supporting  bearings  as  possible. 

If  line-shafting  is  subjected  to  simple  torsion  the  equation 

Mt=^—j^  may  be  used  to  determine  its  diameter.     It  is  frequently 
*  See  Proc.  of  Inst.  of  M.  E.,  1883,  p.  182. 


202  MACHINE  DESIGN. 

convenient  in  such  a  case  to  express  Mt,  the  twisting  moment 
in  inch-pounds,  in  terms  of  the  horse-power  to  be  transmitted, 
H.P.,  and  the  number  of  revolutions  per  minute  of  the  shaft,  N. 
Then,  H. P.  X 33,000X12  =  inch-pounds   of  work   per   minute 

H.P.  H.P. 

---  =  63,024--, 


and  d*  . 

7lfs  N 

If  /,  =  9ooo   Ibs.   per  square   inch   for  mild   steel,    repetitive 
stress  between  o  and  maximum,  one  direction  only, 

~p7 


For  ordinary  line-shaft,  where  there  is  an  average  amount 
of  bending  as  well  as  torsion,  the  common  rule  of  practice  is 


3  H.P. 


For  prime  mover  shafts  and  jack-shafts  the  rules  of  practice 

3,'H.PT         3/fLpT 

for  d  range  from  4.64^——  to  5^—  —  ,  when  the  beanngs  are 

close  to  main  sheaves  or  pulleys. 

In  other  cases  it  is  better  to  compute  the  diameter  by  equa- 
tion (3)  for  combined  torsion  and  flexure,  choosing  the  value 
of  /  according  to  the  material  to  be  used  and  the  conditions  of 
stress  application  and  variation,  i.e.,  in  regard  to  maximum 
and  minimum  repetitive  stresses,  reversals,  and  gradual  or  sudden 
application  of  load. 

If  line-shafts  are  designed  wholly  for  strength,  owing  to 
their  length  there  is  apt  to  be  an  excessive  angular  distortion  &. 
It  is  therefore  desirable  to  design  them  for  stiffness  and  check 
for  strength  afterwards. 


AXLES,  SHAFTS,  AND  SPINDLES.  203 

tf  should  not  exceed  0.075°  per  foot  of  shaft. 

Combining  this  rule  with  the  formula  (5)  for  angular  distortion, 

_MthSo 

&~     ^TF'      » 
JGn 

for  a  round  wrought-iron  or  steel  solid  shaft  this  gives 


V1v~' 

when  d= diameter  of  shaft  in  inches; 

H.P.  =  horse-power  to  be  transmitted; 
•N  =  re  volutions  per  minute  of  shaft. 

Having  determined  the  diameter  which  will  give  sufficient 
stiffness  against  torsion  the  allowable  distance  between  sup- 
porting bearings  must  be  calculated.  The  rule  of  practice  is 
to  limit  the  deflection  to  i/ioo  of  an  inch  to  a  foot  of  length. 

Consider  first  a  bare  shaft  supported  at  both  ends.  There 
are  three  cases : 

ist.  Both  ends  of  the  shaft  are  free  to  take  any  direction. 
2d.   One  end  is  free  and  one  fixed. 
3d.   Both  ends  are  fixed. 
In  each  case  /= length  of  span  in  inches; 
w  =  weight  of  shaft  per  inch; 
y  =  maximum  deflection; 

2  Ay 

—-  =  average  deflection  per  foot  of  length; 

=  1/100  inch; 

2400* 

Each  case  is  that  of  a  uniformly  loaded  beam  with  a  load=w/. 

™/4 
For  case  I,     the  deflection  y  = 


wl* 
For  case  II,    the  deflectioh  y 


204  MACHINE  DESIGN. 

For  case  III,  the  deflection  y=-  -^7. 

3&4/U 

Since    y  = ,    and    for    round   shafting   /=—     and   w  = 

2400  64 

Tld2 

.28 — ,  while  £=29,000,000,  it  follows  that 
4 


(Case  I)  1= 

(Case  II)  /=   So\/~d2; 

(Case  III)  /=  103^. 

When  there  are  loads  due  to  belt  pull,  etc.,  the  deflection 
must  be  determined  in  each  case.  For  ordinary  purposes, 
with  the  average  number  of  pulleys  and  amount  of  belt  pull, 
it  is  safe  to  take  for  loaded  shafts  T%  of  the  value  of  /  deter- 
mined for  bare  shafting  for  the  same  conditions.  Case  II 
corresponds  most  closely  to  ordinary  conditions. 

127.  Critical  Speed  of  Shafts. — It  is  a  practical  impossibility 
to  have  the  center  of  mass  of  a  rotating  shaft  lie  in  the  mechan- 
ical axis  of  rotation.  -  In  the  case  of  a  horizontal  or  inclined 
shaft  it  is  obvious  that  there  will  be  a  deflection,  due  to  its  own 
weight  between  supports  even  if  there  is  no  additional  load, 
which  will  cause  the  center  of  mass  to  lie  below  the  true  axis 
of  rotation.  As  the  shaft  begins  to  rotate,  because  its  center 
of  mass  lies  off  the  axis  of  rotation,  it  will  be  subjected  to  a 
"  centrifugal  force  "  tending  to  make  it  take  a  bowed  form  in 
rotating.  At  first  this  tendency  is  not  sufficient  to  overcome  the 
tendency  to  remain  bent  downward,  but  as  the  speed  increases 
the  "  centrifugal  force  "  increases  until  a  state  of  equilibrium  is 
reached,  at  what  is  called  the  critical  speed,  when  the  centrifugal 
force  will  be  just  sufficient  to  counteract  the  force  tending  to 
deflect  the  shaft  downward  and  the  shaft  will  rotate  in  a  bowed 
form.  It  is  then  said  to  whirl. 

In  the  case  of  vertical  shafts  the  rotating  body  would  have 


AXLES,  SHAFTS,  AND  SPINDLES.  205 

to  be  in  perfect  balance,  which  is  only  theoretically  possible, 
otherwise — and  this  is  always  the  case — the  center  of  mass  lies 
off  the  mechanical  axis.  As  the  speed  increases  there  will  be  the 
same  conflict  between  centrifugal  force,  tending  to  bow  the  shaft 
more  and  more,  and  the  elastic  resistance  of  the  material.  This 
action  continues  until  a  state  of  equilibrium  is  reached  at  which 
the  force  of  the  shaft  deflection  (i.e.,  the  force  which  the  shaft 
is  capable  of  exerting  at  its  mass  center  in  its  effort  to  return  to 
its  original  straight  form)  is  equal  and  opposite  to  the  "  centrif- 
ugal force  "  of  the  mass.  This  will  be  at  the  same  critical  speed 
as  in  the  horizontal  case. 

Theoretically  the  bow  of  the  shaft  becomes  indefinitely  great 
at  the  critical  speed.  Practically  this  is  not  so.  The  speed  may 
be  increased  very  greatly  beyond  this.  But  at  the  critical  speed 
there  will  occur  the  maximum  vibration  of  the  revolving  mass 
and  consequently  of  its  supports.  The  vibrations  are  smaller 
for  both  higher  and  lower  speeds.  The  determination  of  the 
critical  speed  is,  therefore,  a  matter  of  prime  importance  in  the 
design  of  all  high  rotative  speed  machinery. 

Above  the  critical  speed  the  center  of  mass  revolves  inside 
the  bow  of  the  shaft;  and  the  tendency  of  the  rotating  mass  is 
to  rotate  about  an  axis  through  its  own  center  of  gravity  and 
not  about  the  mechanical  axis.  Designers  provide  for  this  either 
by  the  use  of  a  flexible  shaft  (made  just  strong  enough  to  with- 
stand the  deflection  stresses  in  passing  through  the  critical  speed), 
or  by  the  introduction  of  special  bearings  permitting  this  ac- 
commodation and  constructed  to  dampen  the  vibration  effect. 

The  critical  speed  here  spoken  of  is  the  first  or  lowest  critical 
speed,  NI.  There  is  a  series  of  secondary  critical  speeds,  AT, 
of  higher  value  with  diminishing  amplitudes  of  vibration. 

The  mathematical  treatment  is  too  long  to  be  introduced 
here  but  can  be  found  in  its  simplest  form  in  a  paper  by  Mr. 
S.  H.  Weaver  in  the  Jour,  of  the  A.  S.  M.  E.,  June,  1910,  from 


206  MACHINE  DESIGN. 

CRITICAL  SPEED  FORMULAE. 

Weights  in  Pounds,  Dimensions  in  Inches,  Vertical  Shafts  Considered 

Horizontal. 
N,  Ni,  NZ  =  critical  speeds  in  r.  p.  m. 

AI,  A2  =  static  deflections  at  Wi  and  Wz  (shaft  horizontal). 
</  =  diameter  of  shaft  (inches).     £=29,000,000. 


Single   concentrated   Load. 

N      l87'7 

General  Formula  

^-vz: 

i 

1 

7^x87.7^ 
d2  n~ 

Ni  —  387  ooo  —  \  /  — 

i 

E=±^3 

«,    357,000^^^ 
Wia*b* 

Al~  .3EII 

3 

I 

^=1,550,500^2^^ 

AT,  —  787  7  •%/— 

4 

ti-±_-:t::ii-4 

iVl      I07'7    \A! 
A      W3 

Al"48£7 

S 

/        73 
A7         'jQ'?  i-vvi/72  ^  I 

T       f 

7Vi-3°7'00(   5    \^a363 
Vi-i87  7\/— 

j 

|^-.7fbiry4iTtii 

107'7  \A! 
'  .     Wiasb* 

2  lEIl* 

7 

T 

N^s^oo^od.^^ 

A7,  —  TQ7  7  A  /  J 

8 

j^:±.-:^-:±::-i 

57-7  N/A! 

A       Wl13 

Al        IQ2E7 

<f2/     /         / 

r 

77^°°ab  \^1a(3^+&) 

AT,  —  rR7   7  A  /  I 

g<  a  4*---b--*t 

«i     i»7-7  <^— 
A       ^^V/lft) 

Al-I2£//3(3/16) 

ii 

Wi 

JVl=2'337-000T\/l^ 

AT    _  TQ^   n  -«  /    I 

12 

^*  —  —  -*K  L  —  4 

^^__J2  i1  2  ^\ 

^V:-i«7.7^Ai.. 

A        7    W3 
768   £/ 

J3 

AXLES,  SHAFTS,  AND  SPINDLES. 
CRITICAL  SPEED  FORMULA. — Continued. 


207 


T 

*i 

^1  =  387,000**^— 

14 

1  <-      =3 

1~l87'7\A1 
PFi/3 

Al=^ 

15 

^i  and  N2 

Two  Concentrated  Loads 
General  Formulae 

I/K,      Kt\         l/Kt      XA'      4^32 

16, 

^^^VU^^^xWr'W  '  wi^i 

6£7 

17 

/ 

w.           w, 

1    1 

(/-a1-o2)2[/(3^-2ai-202)-(a2-a1)2] 
^i  =  C^/(/-a2)2 

18 

ra     ra 

K                       I                       >) 

^2  =  C—  K/-ai)2 

022 

^      ^(/2-a22-ai2)-a1a2(ai-a2) 

iy 
20 

AS       U 

aiOa 
JVi  and  JV2  =  Substitute  in  Equation  (16) 

d2           I~T 

AT         r  i  ff    inn                           -1  / 

lVl     54°'40°ai(/-2a1)\PF1 

22 

w,           w, 

d2     1         i 

' 

^  548,400  aiMWi(3l_4ai) 

^3 

t±nL,_h3 

N2=  187.7  V^ 
A     ^WH    ,a^ 

24 

Al      6E/  (3      4  ° 

25 

c                   3£/ 

w,            w, 

^^(/i+^-^^+^^^C/i+ax)2 
Kx=  (^  2C[4/2(/1+/2)-(/2+a2)2] 

20 

27 

f^ff-M^^M 

^=(-77)  C[4/i(/i+/2)-(/i+ai)2] 

\a202/ 

7-^                       ^1^2           /^//        1      „     \  //        L/T    "\ 

28, 

AS—     ,      ,  ^(ti"\-ai)(h-\-(ii) 
aibidzbz 

Ni  and  ^2  =  Substitute  in  Equation  (16) 

29 

w                   w. 

r              3EH 

I1     r 

2a262[4/2-(/+a)2] 
Ki-K.-Cfil2     (l+a)2] 

3° 

•21 

|*-a->{*-  -  -6—  Js-  -  ^>—  >f*«^T 
U  j  ,),  f_-_»j 

K3=C(l+a)2 
Ni  and  Nz  =  Substitute  in  Equation  (16) 

32 

208 


MACHINE  DESIGN. 
CRITICAL  SPEED  FORMULA.— Continued. 


1  1 

Afl  =  1  24.3-^  — 

A7                                        J2^  /       T 

33 
34 

"2     2,337,oooa  -y 
•ZV2=  187.7  -y-^- 

35 
36 

37 

C—                    i2s± 

1    I 

39 

40 

fc-*-4^.-b-.«^ 
I*            frj            *f*  "  ""i's"  "**f 

2Vi  and  7V2  =  Substitute  in  Equation  (16) 
TFia262            a/2        8      2x 

Wrf?         '6£//1a'2 

42 

Az"3£/(l      z)         ^E///1 

43 

C3    ^ 

T1         7s 

"1  "2   (ll   1    '2/  —  T6*l  '2 
/Ci  =  l6C/2   (»1~|    '2) 

44 

45 
46 

47 

j^rtj,..^ 

JVi  and  2V2  =  Substitute  in  Equation  (16) 

.0 

^fwA+trv./,/^ 

45 

A"          3#/           i6£/ 

49 

Distributed  Loads 

A  =  Maximum  Static  Deflection 

Total  Load  =W 

JVi  =  2  ,  23  2  ,  5  1  Od  2A/  :JT™ 

50 
Si 

1"           '           -J 

JVi=  4,  760,000  yj     (Shaft  alone) 
N  =  [1,4,  9i  16,  etc.]^i 
'"384  £/ 

52 
53 
54 

AXLE 5,  SHAFTS,  AND  SPINDLES. 
CRITICAL  SPEED  FORMULA. — Continued. 


209 


||  .       Total  Load  =W              ^ 

iuuuiiuuunl 

#1  =  4,979,25°<*2Vf^/~3 

*-«4sVi 

Ni  =  10,616,740  jj     (Shaft  alone) 

tf  =  [i,  2.78,  5.45,  9,  etc/M 
Wl* 

55 
56 

57 
58 
59 

i        i 

3«4£/ 

||        Total  Load  =W 

•iilllliillilllJ 

Ni  =  795,196^  ^-3 

^1  =  167.6^ 

^1  =  1,695,514^-     (Shaft  alone) 

N  =  [i,  6.34,  17.6,  43.6,  etc.jtfi 
TF/3 
'~8£7 

60 
61 
62 
63 
64 

*      '—  ' 

H          Total  Load  =W 

•uiniHumn 

^  =  3,482,715^-^^ 

^i  =  209.7  -y^- 

Ni  =  7,02  1  ,600  —     (Shaft  alone) 

N  =  [i,  3.24,  6.8,  1  1.  6,  etc.]  Ni 
Wl* 

65 
66 

67 
68 
69 

~iSsEI 

which  the  appended  convenient  tables  are  abstracted.  In  addi- 
tion to  these  tables  the  following  formula  *  for  a  multiple  loaded 
shaft  will  be  found  useful. 


NI  =  first  critical  speed  of  the  system ; 
Na  =  first  critical  speed  of  unloaded  shaft ; 
Nb,  Ne= first  critical  speed  of  each  separate  load. 


*  E.  A.  Lof  in  Machinery,  Feb.,  1909. 


CHAPTER  XII. 

JOURNALS,   BEARINGS,   AND   LUBRICATION.* 

128.  General  Discussion  of  Journals  and  Bearings. — Jour- 
nals and  the  bearings  or  boxes  with  which  they  engage  are 
the  elements  used  to  constrain  motion  of  rotation  or  vibration 
about  axes  in  machines.  Journals  are  usually  cylindrical,  but 
may  be  conical,  or,  in  rare  cases,  spherical.  The  design  of 
journals,  as  far  as  size  is  concerned,  is  dictated  by  one  or 
more  of  the  four  following  considerations. 

(1)  To  provide  for  safety  against  rupture  or  excessive  yield- 
ing under  the  applied  forces. 

(2)  To  provide  for  maintenance  of  form. 

(3)  To  provide  for  maintenance  of  lubrication. 

(4)  To  provide  against  overheating. 

To  illustrate  (i),  let  Fig.  122  represent  a  pulley  on  the 
end  of  an  overhanging  shaft  driven  by  a  belt,  ABC.  Rota- 
tion is  as  indicated  by  the  arrow,  and  the  belt  tensions  are  7\ 
and  T2.  The  journal,  /,  engages  with  a  box  or  bearing,  D. 
The  following  stresses  are  induced  in  the  journal:  TORSION, 
measured  by  the  torsional  moment  (T1-T2)r.  FLEXURE, 
measured  by  the  bending  moment  (7\  +  T2)a.  This  assumes 
a  rigid  shaft  or  a  self-adjusting  box.  SHEAR,  resulting  from 
the  force  T1  +  T2.  This  journal  must  therefore  be  so  designed 
that  rupture  or  undue  yielding  shall  no.t  result  from  these 
stresses.  The  method  of  doing  this  is  outlined  in  later  sections 
of  this  chapter, 

To  illustrate  (2),  consider  the  spindle  journals  of  a  grind- 
ing-lathe.  The  forces  applied  are  very  small,  but  the  FORM 

*  See  further,  Vol.  27  Trans.  A.  S.  M.  E.,  pp.  420-505. 

•     210 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


211 


of  the  journals  must  be  maintained  to  insure  accuracy  in  the 
product   of   the    machine.     A    relatively   large    wearing   surface 


FIG.  122. 

is  therefore  necessary,  and  careful  provision  must  be  made  to 
exclude  dust  and  grit.  Journals  whose  maintenance  of  form  is 
of  chief  importance  must  be  designed  from  precedent,  or  accord- 
ing to  the  judgment  of  the  designer.  No  theory  can  lead  to 
correct  proportions.  In  fact  these  proportions  are  eventually 
determined  by  the  process  of  machine  evolution. 

129.  Journal  Friction  and  Lubrication. — Consideration  (3). 
When  two  solid  surfaces  have  relative  motion  under  pressure 
there  will  be  friction  between  them.  The  ratio  of  this  frictional 
resistance,  F,  to  the  normal  pressure,  P,  is  termed  the  coefficient 
of  friction,  //. 

Cylindrical  journals  having  a  diameter =d  inches  and  length 
=  /  inches,  have  a  projected  area  =  d/  square  inches.  In  what 
follows  the  area  of  a  journal  means  its  projected  area.  Thus 

p 

the  pressure  per  square  inch  of  journal  =  />  =  —.     In  machinery 

dl 

there  are  three  kinds  of  friction  to  be  considered. 


212  MACHINE  DESIGN. 

i.  The  surfaces  are  dry. —  In  this  case  the  ordinary  laws 
of  solid  (dry)  friction  apply.  With  very  smooth  and  clean  sur- 
faces cohesion  may  take  place  and  cold  welding  result.  However, 
the  friction  between  most  so-called  unlubricated  surfaces  is  not 
a  case  of  true  friction  between  pure  metals,  but  between  sur- 
faces contaminated  by  atmospheric  agencies,  by  grease,  etc., 
derived  from  handling  or  from  the  material  with  which  the 
surfaces  were  wiped,  or  by  chemically  formed  films  such  as 
oxides,  sulphides,  etc.  In  other  words  the  surfaces  are  partially 
lubricated.  Since  the  extent  of  contamination  varies  it  is  clear 
why  different  experimenters  have  obtained  conflicting  results 
(even  when  using  materials  of  like  character  as  regards  surface 
conditions,  hardness,  etc.)  and  have  deduced  differing  laws. 

For  approximately  clean,  .dry,  metallic  surfaces: 

(a)  The  frictional  resistance  is  approximately  proportional  to 

the  load. 

(b)  The  frictional  resistance  is  slightly  greater  for  large  areas 

and   small   pressures   than  for   small   areas   and   large 
pressures. 

(c)  The  frictional  resistance,   with  the  possible  exception  of 

very  low  speeds,  decreases  as  the  velocity  increases. 

With  regard  to  machinery  these  laws  apply  to  some  friction 
clutches  and  brakes  and  also  to  those  bearings  which  are  not 
lubricated  but  have  the  journal  running  in  bushings  of  graphite 
or  chemically  treated  wood.  Such  bearings  have  been  found  to 
give  very  satisfactory  service  where  the  unit  pressures  are  moderate 
and  where  it  is  either  difficult  to  apply  oil  or  undesirable  to  use 
it,  as  in  textile  manufactures.  These  laws  also  apply  in  the 
starting  of  heavy  machinery  where  the  bearing  surfaces  come  into 
metallic  contact  owing  to  their  having  lain  at  rest  a  sufficient 
time  under  pressure  adequate  to  force  out  the  lubricant,  even 
though  the  natural  action  of  running  tends  to  reintroduce  it  be- 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  213 

tween  them  later.  In  the  case  of  metallic  contacts  the  static 
coefficient  of  friction  of  slightly  contaminated  surfaces  is  the 
highest  value  their  coefficient  of  friction  can  have.  For  the 
ordinary  materials  used  for  journals  and  boxes  static  jj.  ranges 
in  value  from  .14  to  .22. 

In  the  case  of  a  soft  surface  on  a  hard  one,  such  as  a  leather 
belt  on  a  cast-iron  pulley,  these  laws  do  not  hold  strictly.  In 
such  cases  it  has  been  found  *  that  the  coefficient  of  friction 
starting  with  a  static  value  of  say  .12  increases  with  velocity  of 
slip  until  it  reaches  a  maximum  considerably  greater  than  unity 
at  a  speed  of  slip  of  600-700  feet  per  minute. 

2.  The  surfaces  are  partially  lubricated. — Here  the  investigator 
is  confronted  by  conflicting  data,  since  the  experiments  range 
from  nearly  dry,  uncontaminated  surfaces  on  the  one  hand, 
to  completely  lubricated  surfaces  on  the  other.  The  following 
generalizations  for  this  condition  may  be  made : 

(a)  The  coefficient  of  friction  increases  with  increase  of  unit 

pressure. 

(b)  The  coefficient  of _  friction  decreases  with  increase  of  velocity. 

The  value  of  a  will  range  between  its  values  for  dry,  static 
friction  and  for  fluid  friction,  depending  upon  the  conditions. 
Most  machinery  bearings  fall  in  this  category.  Great  care  should 
be  exercised  to  see  that  the  pressure  upon  a  journal  resulting 
from  the  applied  load  be  not  sufficiently  great  or  localized  or 
long  continued  to  squeeze  out  the  lubricant  already  between 
the  surfaces  and  to  prevent  other  lubricant  from  entering  under 
the  conditions  of  speed,  etc.  Metallic  contact,  overheating  and 
abrasion  of  the  surfaces,  even  their  seizing,  may  result.  When 
a  partially  lubricated  journal  is  subjected  to  continuous  pressure 
applied  at  one  point  in  one  direction,  as  for  instance,  a  shaft 

*  Kimball,  Am.  Jour,  of  Science,  1877;  Lanza  and  Lewis,  Trans.  A.  S.  M,  E., 
Vol.  VII. 


214  MACHINE  DESIGN. 

with  a  constant  belt  pull  or  with  a  heavy  fly-wheel  upon  it,  this 
pressure  has  sufficient  time  to  act  and  is  therefore  effective  for 
the  removal  of  oil.  But  if  the  direction  of  the  pressure  is  peri- 
odically reversed  as  at  the  crank-pin  end  of  a  steam-engine  con- 
necting-rod, the  time  of  action  is  less,  the  tendency  to  remove  the 
oil  is  reduced,  and  the  oil  has  opportunity  to  return  between  the 
surfaces.  Hence  a  higher  unit  pressure,  p,  would  be  allowable  in 
the  second  case  than  in  the  first.  If  the  direction  of  relative  motion 
is  also  reversed,  as  at  the  cross-head  pin  of  a  steam-engine,  the 
oil  not  only  has  an  opportunity  to  return  between  the  surfaces, 
but  is  assisted  in  doing  so  by  a  sort  of  pumping  action.  There- 
fore a  still  higher  unit  pressure  is  allowable.  That  practical 
experience  confirms  these  conclusions  can  be  seen  by  reference 
to  Table  XV.  These  conclusions  are  not  applicable  to  the 
cases  of  forced  or  flooded  lubrication  where  there  is  a  copious 
supply  of  oil  between  the  surfaces  in  any  case. 

3.  The  surfaces  are  copiously  lubricated. — The  theory  of 
proper  lubrication  is  to  provide  that  the  bearing  surfaces  shall 
be  separated  always  by  an  unbroken  film  of  lubricant  on  the 
bearing  or  pressure  side ;  there  is  thus  no  metallic  contact  what- 
ever, the  journal  being  fluid-borne.  It  is  obvious  how  this  may 
be  done  by  forced  lubrication  under  sufficient  pressure.  But 
every  continuously  rotating  journal  provided  with  sufficient  oil 
tends  to  surround  itself  with  an  oil-film.  The  extent  to  which 
the  film  is  completely  formed  and  maintained  will  be  found  to 
depend  upon  a  variety  of  factors  of  which  the  chief  are:  the 
difference  in  radius  of  journal  and  bore  of  box,  the  viscosity  of  the 
oil,  the  surface  velocity  of  the  journal,  the  specific  load  or  pres- 
sure per  square  inch  of  projected  area  of  journal,  and  the  tem- 
perature of  the  bearing. 

For  the  process  of  film  formation  see  Fig.  122^4 .  When  a 
journal  under  a  given  load,  P,  is  at  zero  velocity  it  rests  on  a  point 
of  contact  vertically  below  its  load  as  seen  at  a.  As  rotation 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


begins,  the  journal  rolls  upward  on  the  "  on  "  side  of  the  bearing, 
as  shown  at  b,  until  the  angle  (f>  included  between  the  line  of 
application  of  the  load  and  a  radial  line  to  the  point  of  con- 
tact equals  the  "  angle  of  repose  "  or  static  friction  angle  of 
journal  and  bearing  materials  when  slightly  greasy.  This  is 
the  angle  whose  tangent  equals  the  static  coefficient  of  friction 
of  contaminated  surfaces  as  a  maximum.  When  the  journal 


FIG.  122^4. 

has  rolled  to  this  point  it  begins  to  slide.  If  the  velocity  is  very 
low  it  will  continue  to  slip  with  contact  at  this  point  and  the 
ordinary  laws  of  nearly  dry  friction  will  govern.  As  the  speed 
increases  (the  load  P  remaining  constant)  the  conditions  alter. 
Because  of  its  properties  of  adhesion  and  surface  tension,  the 
oil  is  drawn  in  by  the  rotating  journal,  as  illustrated  at  <:,  wedging 
the  journal  completely  away  from  the  bearing  and  moving  the 
point  of  nearest  approach  over  to  the  "  off "  side.  During 
this  period  the  laws  of  partial  lubrication  govern:  the  coefficient 


216  MACHINE  DESIGN. 

of  friction  falls  steadily  as  the  velocity  increases,  and  it  is  higher 
in  value  the  greater  the  unit  pressure,  p,  is. 

By  this  continuous  action  of  the  lubricant,  a  film  (increasing 
in  thickness  with  increasing  velocity  of  journal  surface)  is  formed, 
separating  the  journal  and  bearing.  Coincidently,  shown  in  d, 
the  point  of  nearest  approach  is  moved  toward  point  B  on  the 
horizontal  diameter  on  the  "  off "  side.  The  pressures  in  the 
surrounding  film  are  not  uniform.  A  is  the  point  of  maximum 
pressure  and  lies  just  ahead  of  the  point  of  nearest  approach, 
B,  as  is  to  be  expected  when  the  wedging  action  of  the  oil  before 
the  narrowed  passage  at  B  is  considered.  Beyond  B,  at  C,  is 
the  point  of  minimum  film  pressure  at  which  experiments  show 
that  an  actual  negative  pressure  or  suction  exists. 

As  the  speed  increases  (the  load  remaining  constant)  the 
journal  becomes  less  eccentric  and  the  variations  in  pressure 
around  it  also  become  less.  Both  A  and  C  move  away  from  B% 
If  the  speed  became  infinite  the  journal  would  run  concentric 
with  the  bore  of  the  bearing  and  the  points  of  maximum  and 
minimum  pressures  would  be  vertically  below  and  above  the 
center,  respectively. 

If  the  conditions  shown  at  d  had  been  attained  under  a 
certain  relationship  of  load  and  speed  and  this  speed  now  kept 
constant  while  the  load  were  increased,  the  point  of  nearest  ap- 
proach would  swing  downward  again  to  a  position  about  40° 
from  the  vertical.  With  still  further  increase  of  load  at  this 
constant  speed  the  oil  film  would  be  ruptured  and  the  conditions 
change  again  from  those  of  perfect  to  those  of  imperfect  lubrication. 

But  while  the  condition  of  complete  separation  of  journal 
and  bearing  surfaces  exists,  the  friction  is  fluid  friction — the 
correct  theory  of  which  was  developed  independently  by  Petroff  * 
and  Reynolds. f  In  fluid  friction: 

*  "Neue  Theorie  der  Reibung. ' '     Leipzig,  1887. 
t  Phil.  Trans.,  1886. 


JOURNALS,  BE 4 RINGS,  AND  LUBRICATION.  217 

(a)  The  coefficient  of  friction  varies  directly  as  the  viscosity 
of  the  lubricant. 

(b)  The  coefficient  of  friction  varies  directly  as  the  velocity. 

(c)  The  coefficie.  it  of  friction  varies  inversely  as  the  specific 

pressure. 

(d)  The   coefficient  of  friction  varies  inversely   as   the  mean 

thickness  of  film  on  the  loaded  side  of  the  journal. 

That  is,  j"  =  — . 

py 

T?  is  the  "  coefficient  of  viscosity  "  and  may  be  denned  as 
the  force  necessary  to  move  with  unit  velocity,  one  unit  of  area 
under  unit  pressure,  when  the  two  surfaces  are  separated  by  a 
unit  thickness  of  liquid.  A  study  of  lubricants  shows  that  y 
varies  inversely  as  a  function  of  the  temperature.  On  the  average 
C 

^~(/-32)2' 

V  is  the  journal  surface  speed; 

p  is  the  pressure  per  unit  of  projected  area  of  bearing; 

y  is  the  mean  film  thickness. 

Strictly  this  equation  should  be  kept  in  homogeneous  terms; 
practically  it  is  more  convenient  to  express  V  in  feet  per  minute, 
p  in  pounds  per  square  inch,  y  in  inches,  and  77  in  pounds  per 
square  inch  for  a  speed  of  one  foot  per  minute. 

For  the  maintenance  of  a  perfect  film  the  relation  p= 
may  be  accepted  for  speeds  from  50  to  500  feet  per  minute.  This 
value  is  on  the  safe  side  according  to  the  experiments  of  both 
Stribeck  *  and  Tower,|  although  a  little  larger  than  the  value 
proposed  by  Moore  {  whose  experiments  were  only  carried  up 
to  a  speed  of  140  feet  per  minute. 

Stribeck' s  results  show  the  breaking-down  point  of  the  film 

*  Z.  d.  V.  d.  I.,  1902. 

t  Proc.  Inst.  M.  E.,  1883. 

J  Amer.  Mach.,  1903. 


218  MACHINE  DESIGN. 


at  p  =  2o^v.  Tower's  results  on  a  half-box  give  p= 
to  15  V  F.  The  breaking-down  point  corresponds  to  the  mini- 
mum value  of  the  coefficient  of  friction.  For  speeds  above 
500  ft.  per  minute  10  v  V  gives  too  great  a  value  for  p  unless 
the  bearing  is  artificially  cooled.  Above  500  ft.  per  minute 
p  may  be  taken  =30^7. 

Concerning  y  the  following  facts  hold  : 

(a)  It  is  a  function  of  the  running  fit  allowance.     The  greater 
the  difference  in  radii  of  journal  and  box,  <z,  the  larger  y  has  a 
chance  of  becoming. 

(b)  It  is  a  function  of  p.     The  relation  is  complex.     On  the 
one  hand  y  increases  with  p  because  p  affects  the  bore  by  elastic 
action.     An  increase  of  p  tends  to  force  out  the  box  at  the  point 
of  nearest  approach,  thereby  increasing  the  mean  film  thickness. 
On  the  other  hand  y  decreases  with  increase  of  p,  because  the 
effect  of  increase  of  pressure  on  the  film  must  be  to  lessen  its 
thickness.      The  latter  effect  overbalances  the  former,  hence  y 
varies  inversely  as  a  function  of  p. 

(c)  It  is  a  function  of  V.     As  the  velocity  increases  the  film 
builds  up  and  increases  in  thickness  as  some  function  of  V. 

(d)  It  is  a  function  of  the  temperature,  /,  of  the  bearing. 
As  the  temperature  increases  and  the  "  body  "  of  the  oil  dimin- 
ishes in  consequence,  the  film    for  a  given  pressure  will  be  re- 
duced in  thickness. 

The  final  result  of  the  combined  effects  of  the  various  factors 
becomes  approximately: 

CF* 


where  C  is  a  constant  depending  upon  the  running  fit  allowance 
and  the  viscosity  of  the  lubricant.  For  the  Deutz  engine  oil 
used  by  Stribeck,  and  a  running  fit  allowance  of  .001  inch  per 
inch  of  diameter,  C=.2i8. 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  219 

To  determine  jj.  for  another  lubricant  or  running  fit  allow- 
ance it  is  cnly  necessary  to  vary  C  inversely  as  the  running  fit 
allowance  (e.g.,  double  it  for  a  running  fit  allowance  of  .0005 
inch  per  inch  of  diameter)  and  to  vary  it  directly  as  the  viscosity 
of  the  lubricant.  Measured  with  the  Engler  viscosimeter  the 
oil  used  by  Stribeck  had  a  viscosity  at  86°  F.,  20  times  that  of  water 
at  68°  F.;  and  at  104°  F.,  n  times  that  of  water  at  68°  F. 

This  equation  agrees  exactly  with  Tower's  results  with  sperm 
oil,  and  very  closely  with  his  results  on  mineral  oil,*  remembering 
that  he  has  no  temperature  factor,  since  his  experiments  were 
carried  out  at  approximately  uniform  temperature.  It  also  agrees 
exactly  with  Hirn's  equation  f  except  for  this  temperature  factor. 

The  equation  has  been  thoroughly  checked  on  the  experi- 
mental data  of  Stribeck  J  and  holds  for  speeds  from  50  to  500  ft. 
per  minute. 

For  higher  speed  bearings  it  is  evident  that  as  the  velocity 
increases  to  infinity  y  becomes  equal  to  the  radial  allowance, 
a,  irrespective  of  any  finite  variation  in  p.  The  result  would  be 
that  at  very  high  speeds  the  coefficient  of  friction  would  vary 
directly  as  the  viscosity  at  the  bearing  temperature,  /,  directly 
as  the  velocity,  and  inversely  as  the  pressure, 

rjV 

/.   /£  =  — ,  a  being  constant. 
pa 

This  form  seems  corroborated  by  the   form  of  the  curves  at 
F  =  8oo  ft.  per  minute  in  Stribeck's  Figs.  7  and  17.     Lasche  § 

for    high-speed    bearings    with  copious  lubrication  claims  that 

C 
H  = —  for  V  =  2000,  and  this  is  frequently  quoted  in  the 

(1.2 

form  /*= — —     -  for  high-speed  bearings.     But  some  of  Lasche's 

p(t  ~32) 

*  Thurston,  "  Friction  and  Lost  Work,"  p.  313. 
t  Unwin,  "  Machine  Design,"  1909,  p.  232. 
j  Z.  d.  V.  d.  I.,  1902. 
§  "  Traction  and  Transmission,"  Vol.  6. 


220  MACHINE   DESIGN. 

own  experiments  as  shown  in  his  Fig.  26  do  not  seem  to  justify 
this  conclusion,  since  they  show  /*  still  increasing  with  V  at  speeds 
approximating  5000  ft.  per  minute. 

The  equation  which  satisfies  his  curve   i    (steel  journal  on 
white  metal,  ratio  of  length  to  diameter  2.2)  is  found  to  be 


For    his    curve  4  (nickel  steel  bearing  on  gun  metal,  ratio 
of  length  to  diameter  .42),  the  equation  is 


Since  the  same  oil  was  used  in  each  case  the  difference  in 
the  constants  merely  points  to  double  the  running  fit  allowance 
in  the  second  case  over  that  of  the  first. 

Lasche's  statement  that  /*  varies  as  v  V  up  to  500  ft.  per 
minute,  as  v  V  from  500  to  800  ft.  per  minute,  and  is  inde- 
pendent of  V  above  2000  ft.  per  minute  would  appear  to  require 
modification  as  follows: 

For  speeds  from  50  to  500  ft.  per  minute 


-¥7=v  ....... 

Vp(t  -32) 
approximately, 

(C=.2i8  in  Stribeck's  experiments), 

and  above  500  ft.  per  minute, 


approximately, 

(C=i2  in  Lasche's  experiments.) 

130.  Heating  of  Journals.  —  To   illustrate    (4),    even   if   the 
conditions  are  such  that  the  lubricant  is  retained  between  the 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  221 

rubbing  surfaces,  heating  may  occur.  There  is  always  a  fric- 
tional  resistance  at  the  surface  of  the  journal;  this  resistance 
may  be  reduced  (a)  by  insuring  accuracy  of  form  and  perfec- 
tion of  surface  in  the  journal  and  its  bearings;  (b)  by  insuring 
that  the  journal  and  its  bearings  are  in  contact,  except  for  the 
film  of  oil,  throughout  their  entire  surface,  by  means  of  rigidity 
of  framing  or  self-adjusting  boxes,  as  the  case  may  demand; 
(c)  by  selecting  a  suitable  lubricant  to  meet  the  conditions  and 
maintaining  the  supply  to  the  bearing  surfaces.  By  these 
means  the  friction  may  be  reduced  to  a  very  low  value,  but  it 
cannot  be  reduced  to  zero. 

There  must  be  some  frictional  resistance,  and  it  is  always 
converting  mechanical  energy  into  heat.  This  heat  raises  the 
temperature  of  the  journal  and  its  bearing.  If  the  heat  thus 
generated  is  conducted  and  radiated  away  as  fast  as  it  is  gener- 
ated, the  box  remains  at  a  constant  low  temperature.  If,  how- 
ever, the  heat  is  generated  faster  than  it  can  be  disposed  of, 
the  temperature  of  the  box  rises  till  its  capacity  to  radiate  heat 
is  increased  by  the  increased  difference  of  temperature  of  the 
box  and  the  surrounding  air,  so  that  it  is  able  to  dispose  of 
the  heat  as  fast  as  it  is  generated.  This  temperature,  necessary 
to  establish  the  equilibrium  of  heat  generation  and  disposal, 
may  under  certain  conditions  be  high  enough  to  destroy  the 
lubricant  or  even  to  melt  out  a  babbitt-metal  box-lining.  Sup- 
pose now  that  a  journal  is  running  under  certain  conditions 
of  pressure  and  surface  velocity,  and  that  it  remains  entirely 
cool.  Suppose  next  that,  while  all  other  conditions  are  kept 
exactly  the  same,  the  velocity  is  increased.  All  modern  experi- 
ments on  the  friction  in  journals  show  that  the  coefficient  of 
friction  increases  with  the  increase  of  velocity  of  rubbing  sur- 

P2 

face  (at  speeds  above  -    -    feet  per  minute).     Therefore  the  in- 
400 

crease  in  velocity  would  increase  the  frictional  resistance  at  the 


322  MACHINE  DESIGN. 

surface  of  the  journal,  and  the  space  through  which  this  resist- 
ance acts  would  be  greater  in  proportion  to  the  increase  in 
velocity.  The  work  of  the  friction  at  the  surface  of  the  journal 
is  therefore  increased  because  both  the  force  and  the  space 
factors  are  increased.  It  is  this  work  of  friction  which  has 
been  so  increased,  that  produces  the  heat  which  tends  to  raise 
the  temperature  of  the  journal  and  its  box.  The  rate  of  gen- 
eration of  heat  has  therefore  been  increased  by  the  increase  in 
velocity,  but  the  box  has  not  been  changed  in  any  way,  and 
therefore  its  capacity  for  disposing  of  heat  is  the  same  as  it 
was  before,  and  hence  the  tendency  of  the  journal  and  its  bear- 
ing to  heat  is  greater  than  it  was  'bef ore  the  increase  in  velocity. 
Some  change  in  the  proportions  of  the  journal  must  be  made 
in  order  to  keep  the  tendency  to  heat  the  same  as  it  was  before 
the  increase  in  velocity.  If  the  diameter  of  the  journal  be 
increased,  the  radiating  surface  of  the  box  will  be  proportion- 
ately increased.  But  the  space  factor  of  the  friction  will  be 
increased  in  the  same  proportion,  and  therefore  it  will  be  appar- 
ent that  this  change  has  not  affected  the  relation  of  the  rate 
of  generation  of  heat  to  the  disposal  of  it.  But  if  the  length 
of  the  journal  be  increased,  the  unit  pressure  is  decreased,  which 
tends  to  decrease  the  coefficient  of  friction,  while  the  increase 
of  velocity  tends  to  increase  it.  Due  to  the  combined  change 
the  coefficient  of  friction  may  be  slightly  increased  or  decreased, 
the  velocity  is  increased,  and  the  total  friction  will  be  slightly 
increased,  while  the  radiating  surface  of  the  box  is  increased 
in  greater  proportion  and  the  tendency  of  the  box  to  heat  is 
reduced.  If  the  diameter  of  the  journal  is  reduced  coincidently 
with  increasing  its  length,  so  that  the  unit  pressure  remains  the 
same  both  before  and  after  the  change,  the  velocity  may  be 
reduced,  as  may  also  the  coefficient  of  friction,  and  therefore 
the  friction  work  may  be  doubly  reduced  while  the  radiating 
surface,  which  will  be  proportional  to  the  same  projected  area 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  223 

in  both  cases,  will  remain  the  same.  The  tendency  to  heat 
will  therefore  be  reduced.  If,  therefore,  the  conditions  are  such 
that  the  tendency  to  heat  in  a  journal,  because  of  the  work  of  the 
friction  at  its  surface,  is  the  vital  point  in  design,  it  will  be  clear 
that  the  ratio  of  the  length  of  the  journal  to  the  diameter  is  dic- 
tated by  it.  The  reason  why  high-speed  journals  have  greater 
length  in  proportion  to  their  diameter  than  low-speed  journals 
will  now  be  apparent. 

If  v  equals  the  velocity  of  journal  in  feet  per  second,  p  being 
the  pressure  per  square  inch  of  projected  area  and  u.  the  coeffi- 
cient of  friction,  the  energy  transformed  into  heat  will  be  upv 
ft.-lbs.  per  second  per  square  inch  of  projected  area.  This 
energy  should  be  dissipated  (by  radiation  and  conduction)  through 
a  surface  which  bears  a  relationship  to  the  projected  area  depend- 
ing upon  the  thickness  of  the  shell  and  other  portions  of  the 
bearing.  In  order  to  have  equilibrium  of  heat  generation  and 
heat  dissipation  a  certain  temperature,  /,  in  excess  of  atmospheric 
temperature,  /0,  must  be  attained.  For  a  ring-oiling  bearing 
whose  upper  radiating  surface  was  about  double  that  of  the 
journal  under  running  conditions,  Lasche  found 


3300 

and  for  a  bearing  with  heavy  shells  of  the  turbo-dynamo  type 
and  upper  radiating  surface  about  three  times  that  of  the  journal 


In  both  cases  /  was  about  2.5^. 

Stribeck's  results  on  a  Sellar's  bearing,  I  =3.3(1,  when  similarly 
analyzed  show 


2750 


224 


MACHINE  DESIGN. 


for  speeds  up  to  500  feet  per  minute,  and  an  apparently  more 
rapid  radiation  still  when  /  -/0  exceeded  80°.  This  checks  very 
closely  with  Lasche's  results. 

For    a    shorter    Magnolia    metal    bearing,    l=2d,    Stribeck's 
results  give  a  much  greater  coefficient  of  heat  dispersion, 


but  he  warns  against  their  reliability. 

It  seems  safe  to  assume,  where  there  is  no  extraneous  cooling 
device  such  as  oil  or  water  circulation  or  fan  cooling,  that 


for  bearings  of  ordinary  proportions.  Thick  shells  and  heavy 
masses  in  the  bearing  will  increase  its  heat-  radiating  capacity 
and  thin  shells  and  light  masses  lower  it. 

The  nearness  of  rotating  masses  acting  as  air  fans  will  increase 
the  heat-dissipating  capacity.  Dead  air  spaces  around  the 
shells  decrease  it  greatly.  In  case  the  ordinary  proportions  of 
the  journal  give  too  high  a  value  to  /,  oil-circulation  or  even 
water-jacketing  systems  may  be  used  to  dispose  of  the  excess 
heat.  A  value  of  /  up  to  200°  is  quite  safe  with  ordinary  lubri- 
cants.* 

131.  Journal  Design  by  Heat  Balance.  —  (a)  For  velocities 
up  to  500  fi.  per  minute,  copious  lubrication. 

From  equation  (i) 

CV* 
^l^t^)' 

From  equation  (3) 


2750 


*  Dewrance,  Proc.  Inst.  C.  E..  1896. 


JOURNALS,  BEARINGS,  .AND  LUBRICATION.  22$ 


2750 

and 

CV*        >F/ 


#*(/-32)6o  2750 

or 


60  2750 

As  previously  explained,  C=.2i8  for  Deutz  engine  oil  and 
a  running  fit  allowance  of  .001  inch  per  inch  of  diameter, 


(4) 


With  known  values  of  V  and  p,  and  assumed  value  of  /0,  the  room 
temperature,  this  can  be  solved  by  taking  trial  values  of  /  until 
one  is  found  to  satisfy  the  equation.  For  other  lubricants  and 
running  fit  allowance  C  should  be  varied  as  explained  in  sec.  129. 

Let   p=io^V    ...     (5)  (page  2 17); 

'.      *-£.',.   (6); 

P  =  total   load   on   bearing   in   pounds.     If  load   varies 
throughout  cycle,  P  =  mean  load  during  revolution ; 
/=length  of  journal,  inches; 
d= diameter  of  journal,  inches; 

oc  =  -,  assumed,  see  Table  XVI; 
a 

TV  — revolutions  per  minute. 
P,  N,  and  x  are  given. 


226  MACHINE  DESIGN. 

From  (5),  (6),  and  (7), 
P 

Squaring 

P2 


12 


Multiplying  both  sides  by  d2, 
d*  =  ' 


I0) 


P,  AT",  and  x  being  known,  solve  for  d. 

From  ix?  =  -  ,  /  =  dx.     Solve  for  /. 
d 

p 
From  (6)  -Tj  =  P-     Determine  p. 

From  (7)  V  =  *~  ~.     Solve  for  V. 

12 

Since 

.,,..""  .......     (5) 


Substituting  in  (4), 


31.6 
Solve  this  for  /. 

If  /  has  too  high  a  value  (200°  F.  is  a  good  maximum  value) 
and  neither  less  viscid  oil  nor  greater  running  fit  allowance  may 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


227 


be  used,  a  new  value  of  x  giving  a  greater  ratio  of  length  to 
diameter  must  be  tried;   or  means  provided  to   use  extra  heavy 
masses  in  the  shells  and  bearing  or  to  cool  the  latter  artificially. 
The  following  table  is  based  on  an  assumption  of  /o  =  68°  F. : 


t 

F* 

V 

78° 

2950 

96 

88 

5380 

136 

98 

8850 

180 

1  08 

13600 

231 

118 

19700 

285 

,  128 

27400 

344 

138 

36700 

406 

148 

48500 

476* 

*  At  this  point  the  radiation  curve  departs  from  the  equation  chosen,  allowing  higher 
values  of  V.     See  section  130. 

Having  determined  p,  V,  and  /,  solve  for  p  in  equation  (i), 


re -3*) 

This  can  be  used  to  determine  the  efficiency  of  the  bearing 
which  equals:  (Total  energy  received  per  minute  at  journal  in 
foot-pounds  -fiPV)-T- Total  energy  received  per  minute  at  journal 
in  foot-pounds. 

The  rate  of  heat  generation  per  square  inch  of  bearing  area 
in  foot-pounds  per  second  =  upv. 

[The  foregoing  method  may  also  be  used  in  a  modified  form 
for  a  given  maximum  allowable  value  of  /.  Given  also  P,  N, 
and  x. 

Solve  for  d,  /,  p,  and  V  as  above. 

Compute  fi  from  u  = 
Find  upv. 

If   this   exceeds ,    extraneous   means   of    cooling 

2750 


^ . 


228  MACHINE   DESIGN. 

should  be  provided,  it  being  assumed  that  the  running  fit  allowance 
and  lubricant  may  not  be  changed. 
Two  methods  may  be  used : 

(1)  Increase  the  thickness  of  the    bearing  (above  -)  in  the 

ratio  that  the  excess  bears  to — .      This  provides  the 

2750 

additional  radiating  surface  necessary. 

(2)  Compute,    from    its    specific    heat    and    the    permissible 
range  of  entering  and  leaving  temperature,  the  quantity  of  oil 
or  water  which  must  be  circulated  per  second  through  a  jacket 
arrangement  to  carry  off  the  surplus  heat  generated.] 

(b)  For  speeds  above  500  feet  per  minute,  copious  lubrication. 

.     .     (12)         Page  218. 
P         skdN 


12 
Cubing, 

P3 


7069^/3' 


d-V—r  ^ (J4) 


Solve  (14)  for  d;  P,  N,  and  x  being  given. 
Solve  l=dx  for  /. 


JOURNALS,  BEARINGS,  AND    LUBRICATION. 


229 


Determine  p,  from  P  =  J-J- 
From  (2),  (page  220) 

From  (3),  (page  224) 


P(*  -32) 


2750 


60          2750 


Substituting  from  (2) 


p(t-$2)    60  2750 


550 


Table,  when  /0  =  68°F. 


/ 

V' 

V 

148° 

2800 

572 

158 

3670 

711 

168 

4500 

836 

178 

5600 

997 

1  88 

6800 

1164 

198 

8170 

1350 

208 

9800 

1560 

The  foregoing  method  of  design  was  suggested  by  the  papers 
of  Mr.  Axel  Pedersen  in  the  American  Machinist,  1913  and  1914. 

*  This  equation  gives  rather  higher  values  to  t  than  will  probably  be  attained 
by  bearings  of  ordinary  proportions  using  customary  lubricants.  In  other  words, 
higher  values  of  V  than  these  may  be  expected  in  practice  at  these  temperatures. 
The  equation  is  on  the  safe  side. 


230 


MACHINE   DESIGN. 


TABLE  XV. —  CYLINDRICAL  JOURNAL  PRESSURES  FROM  PRACTICE 


Kind  of  Bearing. 


Pressure  in  Ibs. 
per  sq.  in.  of 
projected  area. 


Motion  intermittent,  direction  of  load  reversing,  slow  speed. 
.  Crank  pins  of  shearing  and  punching  machines,  presses  etc. 


3000-7000' 


Motion  an  oscillation,  direction  of  load  reversing 

Locomotive  cross-head  pins 3000-4000 

Gas  engine  cross-head  pins 2000-3000! 

Air  compressor  cross-head  pins 400-1350 

Slow-speed  stationary  engine  cross-head  pins 1000-1860 

High-speed  stationary  engine  cross-head  pins 910-1675 

Motion  a  rotation,  direction  of  load  changing. 

Locomotive  crank  pins 1400-1700 

Gas  engine  crank  pins iooo-2ooof 

Air-compressor  crank  pins 250-  850 

Marine  engine  crank  pins 400-  500 

Slow-speed  stationary  engine  crank  pins 870-1550 

High-speed  stationary  engine  crank  pins  (center  crank) 250-  600 

High-speed  stationary  engine  crank  pins  (side  crank) 900-1500 

Eccentric  sheaves 80-  100 

Motion  a  rotation,  direction  of  load  nearly  constant. 

Merchant  marine  engine,  main  bearings 200-  350 

Naval  marine  engine,  main  bearings 275-  400 

Slow-pumping  engine,  main  bearings 600 

Slow-speed  stationary  engine,  main  bearings 200-  300 

High-speed  stationary  engine,  main  bearings 180-  240 

Gas  engine,  main  bearings 500-  7oof 

Air  compressor,  main  bearings JS0"  25° 

Car  axle  journals 300-  600 

Locomotive  and  tender  axle  journals 400-  550 

Line  shafts  on  bronze  or  babbitt 100-  150 

Steel  shaft  on  lignum  vitse,  water  lubrication 350 

Practice  of  Gen'l.  Elec.  Co.J 
Ring-oiling  or  other  copious  lubrication. 
Velocity  of  journal,  ft.  per  minute  =  V: 

50-  100  p=7\/V 

100-2000  p=i$.6\/V 

2000-3000  P=3° 

3000-4000  P=44    V 'V 


*  In  Vol.  27,  Trans.  A.  S.  M.  E.,  pp.  496-497,  Mr.  Oberlin  Smith  gives  examples 
of  journal  pressures  in  presses  running  as  high  as  20,000  pounds  per  square  inch 
on  hardened  steel  toggle  pins;  and  7000  pounds  per  square  inch,  at  a  surface  speed 
of  140  feet  per  minute,  against  the  cast-iron  pitman  driving  the  ram.  The  journal 
pressure  of  the  main  shaft  of  the  second  press  was  2400  pounds  per  square  inch. 

f  Based  on  maximum  explosion  pressure. 

t  Data  from  other  sources  indicate  that  these  values  could  be  increased  con- 
siderably with  safety. 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


231 


Departure  from  his  method,  based  upon  different  conclusions 
regarding  values  of  //,  p,  etc.,  are  made  here,  however. 

The  method  given  under  (a)  may  be  applied  to  these  bearings 
also,  to  design  for  a  maximum  given  value  of  /. 

132.  Allowable  Bearing  Pressure. — Table  XV  on  p.  230,  based 
upon  current  practice,  may  be  used  as  a  guide  by  the  designer. 
The  value  to  be  used  in  each  case  is  a  matter  of  judgment.  The 
allowable  pressure  depends,  among  other  items,  upon  the  grade 
of  workmanship  expected  as  shown  in  the  fit  and  surface  con- 
ditions of  the  journal  and  box. 

133-  Journal  Proportions. — Customary  proportions  of  journals 
may  be  seen  in  the  following  table  compiled  from  current  practice: 

TABLE  XVI. 


Value  of  4 
a 

Minimum. 

Maximum. 

Average. 

Main  bearings,  marine  engines 

I 

I    e 

Main  bearings,  center-crank,  high-speed  engine  .  . 
Main  bearings,  side-crank,  slow-speed  engine.  .  .  . 
Main  bearings,  gas  engines  

2 

i-7 

3 

2.1 

2    2 

i.  9 

2    2Z      • 

Crank  pins,  marine  engines  

i 

I    r 

Crank  pins  high-speed  engines 

Crank  pins,  slow-speed  engines 

I 

Crank  pins,  gas  engines 

I    7 

A 

Cross-head  pin,  high-speed  engines  

i 

2 

2$ 

Cross-head  pin,  slow-speed  engines  

i 

I    ir 

Cross-head  pin,  gas  engines  

I     (T 

7$ 

Fixed  bearings  shafting 

2 

Self-adjusting  bearings,  shafting 

Generator  and  motor  bearings  

2 

Machine  tool  bearings  

2 

134.  Materials  to  be  Used. — Regarding  the  materials  of 
journals  and  their  boxes  the  following  general  statements  may 
be  made.  It  must  be  borne  in  mind  that  the  terms  bab- 
bitt, brass,  and  bronze  cover  wide  ranges  of  alloys  of  varying 
values. 

Cast  iron,  wrought  iron,  soft  steel,  and  hard  steel  will  all  run 


232  MACHINE    DESIGN. 

well  at  almost  any  speed  on  babbitt  metal.  The  pressure  per 
square  inch  which  an  ordinary  babbitt  bearing  will  stand  when 
running  cool  (i.e.,  at  very  slow  speed),  before-  being  squeezed 
out,  has  been  found  to  be  something  over  2000  Ibs.* 

Cast  iron,  wrought  iron,  soft  steel,  and  hard  steel  will  all 
run  well  on  brass  and  bronze.  Brass  and  bronze  of  ordinary 
compositions  will  carry  5000  Ibs.  per  square  inch  without  suffer- 
ing destruction.  Bronze,  however,  is  much  better  than  brass. 

Cast  iron  will  run  on  cast  iron  where,  owing  to  large  bear- 
ing surfaces,  the  unit  pressure  is  light.  Where  the  pressure  and 
speed  are  high,  as  in  engine-journals,  this  will  not  work.f 

In  the  same  way  steel  will  run  on  cast  iron  even  at  high  speeds 
if  the  pressure  is  light.  It  has  been  found  that  steel  will  not 
run  on  cast  iron  in  engine- journals.! 

Wrought  iron,  soft  steel,  and  hard  steel  will  all  run  on  hard 
steel. 

Steel  under  steel  if  hardened  and  polished  will  run  under  as 
high  a  pressure  as  50,000  Ibs.  per  square  inch. 

135.  Calculation  of  Journals  for  Strength. — Journals  gener- 
ally form  parts  of  axles  on  shafts,  and  the  calculation  of  their 
diameter  for  strength  becomes  part  of  the  calculation  of  the 
shaft.  The  principles  have  been  developed  at  length  in  the 
preceding  chapter  and  need  not  be  repeated  here. 

If  the  journal  is  so  held  that  it  may  be  considered  as  sub- 
jected to  pure  shearing  stress,  like  the  crank-pin  of  a  center- 
crank  engine,  then 

M-P, 

in  which    P  =  total  maximum  load ; 

A  =  total  area  subjected  to  stress; 

fs  =  safe  shearing  stress  for  the  conditions. 

*  C.  F.  Porter,  Trans.  A.  S.  M.  E.,  Vol.  Ill,  p.  227. 
t  Trans.  A.  S.  M.  E.,  Vol.  VI,  pp.  853-854. 


JOURNALS,  BEARINGS,  AND   LUBRICATION.  233 

For  a  journal  subjected  to  a  pure  bending  moment, 


which  becomes  Pl= for  a  solid  circular  shaft.     Pl  =  bending 

moment,  /  =  safe  unit  stress,  and  r  =  radius  of  shaft.     This  can 
readily  be  solved  for  r. 
If  the  journal  be  hollow, 

Pi_Mr£^>t 


r\  being  the  external  and  f2  the  internal  diameter. 

For  combined  bending  and  twisting  such  as  the  main  journal 
of  a  side-crank  engine  is  subjected  to,  the  expression  for  a  solid 
journal  is 

/—  = 
4 

For  a  hollow  circular  section 


Mb  being  the  bending  moment  and  M  t  the  twisting  moment. 

In  general  it  will  be  found  that  journals  proportioned  for 
strength  merely  will  not  have  sufficient  area  to  prevent  heating, 
so  this  item  must  not  be  overlooked. 

136.  Problem.  —  Design  the  main  journal  of    a    side-crank 
low-speed  engine. 
Diameter  of  cylinder  =  16  ins. 
Length  of  stroke         =36  ins. 
Net  forward  pressure  =  100  Ibs.  per  square  inch  of  piston  area. 

Suppose  the  engine  capable  of  carrying  full  pressure  to  half- 
stroke. 


234 


MACHINE  DESIGN. 


The  area  of  piston  =201.06  square  inches. 

.'.  total  net  forward  pressure  =  20,106  Ibs. 

At  point  of  maximum  torsional  effect,  which  corresponds  to 
the  position  of  maximum  velocity  of  piston,  no  energy  is  used 
in  accelerating  reciprocating  parts,  and 

FpVp=Fcvc', 

Fp  =  net  forward  force  on  piston ; 
vp  =  velocity  of  piston; 
Fc=  force  on  crank; 
vc  =  velocity  of  crank. 

Since  vc  is  less  than  vp  for  this  position,  Fc  is  greater  than 

p  v 

FP,  since  Fe=-*-*-. 

Assuming  a  connecting-rod  length  equal  to  five  and  a  half 
crank  lengths  gives  (Appendix)  Fc  =  2o,$oo  Ibs. 

Since  the  crank  length  is  18  inches,  and  at  this  position  the 
crank  and  connecting-rod  are  nearly  at  a  right  angle  with 
each  other,  there  is  a  twisting  moment  at  the  journal  equal  to 

Jf i  =  20, 500X18— 369,000  inch-lbs. 

There  is  also  a  bending  moment  equal  to  20,500  X  the  dis- 
tance from  center  of  crank-pin  to  center  of  main  journal.  In 

most  cases  this  distance  must  be 
assumed;  for,  although  the  length 
of  the  crank-pin  and  the  thickness 
of  the  crank  may  be  known,  the 
length  of  the  main  journal  is  un- 
known, since  this  length  and  the 
journal  diameter  are  the  very 
dimensions  sought.  Assume  then 
that  the  crank-pin  is  6  inches  long, 
the  crank  3  inches  thick,  and  the  middle  of  main  journal  6 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  235 

inches  from  the  inner  face  of  crank  as  shown  in  Fig.  123. 
This  will  give  12  inches  as  the  lever-arm;  .'.  the  bending  moment, 
Mb,  =  20,500X12  =  246,000  inch-lbs. 

The  equivalent   bending  moment    to  the   combined  actual 
bending  and  twisting  moments 

=Meb  _  _ 

=  0.35  X  246,000  +  o.65\/246ooo2  +  3690002 
=  374>375  inch-lbs. 

But 


4 


a          4X374375 


For  a  main  shaft  like  this  /  may  be  taken  =  12,000  Ibs. 
per  square  inch  for  steel. 

.  3/4X374375 

M  7TXI2000 

=  3.41  inches; 

.*.  diameter  of  journal  =  2X3.4i=6.82,  say  7  inches. 

The  length  according  to  practice  would  be  about  twice  this 
diameter,*  or  14  inches.  This  would  give  a  projected  area  of 
98  square  inches  and  a  pressure  of  something  over  200  Ibs. 
per  square  inch  of  bearing  due  to  steam-pressure  alone. 

To  get  the  actual  maximum  pressure  on  the  journal  it 
would  be  necessary  to  know  the  weight  of  the  shaft,  flywheel, 
and  other  attached  parts,  and  properly  combine  the  pressure 
due  to  these  with  the  pressure  due  to  the  steam. 

The  rough  rule  of  practice  for  Corliss  engines  is  to  make 
the  diameter  of  main  journal  equal  to  one  half  the  diameter 
of  the  cylinder. 

*   See  Table  XVI,  p.  231. 


236  MACHINE  DESIGN. 

137.  Problem.  —  Design  the  crank-pin  for  the  same  engine. 
It  will  be  found  that  the  crank-pin  must  be  designed  with  refer- 
ence to  maintaining  lubrication,  and  that  it  will  have  an  excess 
of  strength. 

Allowing  1  200  Ibs.  per  square  inch  of  area,*  and  noting 
from  the  table  that  the  average  practice  for  this  type  of  engine 
is  to  make  the  length  of  the  pin  =  i.iXthe  diameter,  j  it  fol- 
lows that 


1200 

but  l=i.  id', 

,9     20500 


1200 

and  d  =  4  inches,  nearly; 

.*.  /=  i.i  X4  =  4j  inches,  say. 

Checking  this  for  strength,   considering  the  pin  subjected 

/ 
to  a  bending  moment  P-,  we  write 


2       4 
P  =  20500  Ibs., 

-  =  —  =  2.25  inches; 
2      2 

r=2  inches, 

/  =  stress  in  outer  fiber; 

4X20500X2.25  . 

.*.  /  =  -  '—z  -  1  =7300  Ibs.  per  square  inch; 

7T  X  O 

which  is,  of  course,  a  perfectly  safe  value  for  wrought  iron  or 
steel. 

*  See  Table  XV,  p.  230.  f  See  Table  XVI,  p.  231. 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


237 


138.  Problem.  —  Design  the  cross-head  pin  for  the  same 
engine.  This  pin  also  should  be  designed  for  maintaining 
lubrication.  Allowing  1400  Ibs.  per  square  inch  as  the  per- 
missible pressure  on  the  journal,*  and  noting  that  the  length 
may  be  taken  as  1.3  times  the  diameter  from  average  practice* 
gives 

20500 
, 
1400 


,, 
dif 


and 


1400 

^  =  3l  inches; 
.'.  /  =   inches. 


Checking  this  for  strength  it  is  evident  that  the  only  way 


FIG.  124. 


20500 


this  pin  can  fail  is  by  shearing  on  two  surfaces,  A-B  and  D-E 
(see  Fig.  124). 


per  square 


This  leaves  so  great  a  margin  of  safety  that  some  manu- 
facturers make  the  cross-head  pin  of  two  parts,  an  inner  pin 
of  soft,  resilient  material,  sufficiently  large  to  resist  the  shear- 
ing stress,  and  an  outer  hard-steel  bushing  which  surrounds 
the  soft  pin,  but  is  not  allowed  to  turn  on  it.  The  nature  of 


*  See  Table  XV,  p.  230. 


f  See  Table  XVI,  p.  231. 


MACHINE  DESIGN. 


the  forces  acting  on  a  cross-head  pin  tend  to  wear  it  to  an  oval 
cross-section.  As  such  wear  takes  place  the  bushing  can  readily 
be  given  a  quarter  turn  and  clamped  in  the  new  position. 
(See  Fig.  124.) 

139.  Thrust-journals. — When  a  rotating  machine  part  is 
subjected  to  pressure  parallel  to  the  axis  of  rotation,  "mean^ 
must  be  provided  for  the  safe  resistance  of  that  pressure.  In 
the  case  of  vertical  shafts  the  pressure  is  due  to  the  weight 
of  the  shaft  and  its  attached  parts,  as  the  shafts  of  turbine 
water-wheels  that  rotate  about  vertical  axes.  In  other  cases 
the  pressure  is  due  to  the  working  force,  as  the  shafts  of  pro- 
peller-wheels, the  spindles  of  chucking-lathes,  etc.  The  end- 
thrusts  of  vertical  shafts  are  very  often  resisted  by  the  "squared- 
up  "  end  of  the  shaft.  This  is  inserted  in  a  bronze  or  brass 
"bush,"  which  embraces  it  to  prevent  lateral  motion,  as  in 
Fig.  125.  If  the  pressure  be  too  great,  the  end  of  the  shaft 
may  be  enlarged  so  as  to  increase  the  bearing  surface,  thereby 
reducing  the  pressure  per  square  inch.  This  enlargement 


B 

FIG.  126. 


FIG.  125. 

must  be  within  narrow  limits,  however.  (See  Fig.  126.)  AB  is 
the  axis  of  rotation,  and  ACD  is  the  rotating  part,  its  bear- 
ing being  enlarged  at  CD.  Let  the  conditions  of  wear  be  con- 
sidered. The  velocity  of  rubbing  surface  varies  from  zero 
at  the  axis  to  a  maximum  at  C  and  D.  It  has  been  seen  that 
the  increase  of  the  velocity  of  rubbing  surface  increases  the 
work  of  the  friction,  and  therefore  the  tendency  to  wear.  From 
this  it  will  be  seen  that  the  tendency  to  wear  increases  from 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


239 


the  center  to  the  circumference  of  this  "  radial  bearing,"  and 
that,  after  the  bearing  has  run  for  a  while,  the  pressure  will  be 
localized  near  the  center,  and  heating  and  abrasion  may  result. 
Because  of  low  velocity  at  the  center  it  becomes  difficult  to  main- 
tain the  oil  film  there,  which  also  adds  to  this  local  danger.  For 
these  reasons,  where  there  is  a  heavy  load  to  be  borne,  the  bearing 
is  usually  divided  up  into  several  parts,  the  result  being  what 
is  known  as  a  "  collar  thrust-bearing,"  as  shown  in  Fig.  127. 


FIG.  127. 

By  the  increase  in  the  number  of  collars,  the  bearing  surface 
may  be  increased  without  increasing  the  tendency  to  unequal 
wear.  The  radial  dimension  of  the  bearing  is  kept  as  small 
as  is  consistent  with  the  other  considerations  of  the  design. 
If  dw  =  mean  diameter  of  collar,  its  radial  width  may  be  made 
J  to  T3¥  dm;  and  the  axial  thickness,  f  to  J  this  width. 

It  is  found  that  the  "  tractrix,"  the  curve  of  constant  tan- 
gent, gives  the  same  work  of  friction,  and  hence  the  same  ten- 
dency to  wear  in  the  direction  of  the  axis  of  rotation,  for  all  parts 
of  the  wearing  surface.  (See  "  Church's  Mechanics,"  page  181.) 

This  has  been  very  incorrectly  termed  the  "  anti-friction " 
thrust-bearing.  This  is  far  from  being  the  case.  The  friction 
work  for  this  and  all  conical  thrust-bearings  can  be  shown 
readily  to  be  excessive.  Their  one  advantage  is  that  they  are 
easily  adjustable.  In  general  they  are  to  be  avoided. 

The  pressure  that  is  allowable  per  square  inch  of  projected 
area  of  bearing  surface  varies  in  thrust-bearings  with  several 


240  MACHINE  DESIGN. 

conditions,  as  it  does  in  journals  subjected  to  pressure  at  right 
angles  to  the  axis.*  Thus,  in  the  pivots  of  turntables,  swing 
bridges,  cranes,  and  the  like,  the  movement  is  slow  and  never 
continuous,  often  being  reversed;  and  also  the  conditions  are 
such  that  "  bath  lubrication  "  may  be  used,  and  the  allowable 
unit  pressure  is  very  high — equal  often  to  1500  Ibs.  per  square 
inch,  and  in  some  cases  greatly  exceeding  that  value. 

Tower's  investigation  of  a  water-cooled  collar  bearing  showed 
a  maximum  value  of  p  of  75  Ibs.  per  square  inch  of  net  collar 
area  at  a  mean  collar  speed  of  440  ft.  per  minute,  increasing  to 
90  Ibs.  per  square  inch  at  a  speed  of  170  ft.  per  minute. 

The  lowest  values  of  the  coefficient  of  friction,  /£,  were  obtained 
when  p=  $^V.  At  this  relation  of  p  to  V  they  ranged  from  .0286 
to  .0348  at  p  =  6o  and  82.5  respectively.  When  this  relation  was 
departed  from,  /*  increased  in  value.  When  p  was  15  and  V  was 
440,  jj.  went  up  to  .0646.  Ordinarily  collar  bearings  are  limited 
to  a  pressure  of  50  to  60  Ibs.  per  square  inch  and  a  coefficient 
of  friction  of  .035  may  be  expected  under  customary  conditions 
of  running. 

If  a  single  collar  is  used  with  a  mean  diameter  =  dm  and  a 
radial  width  =  .i$dm,  the  following  equations  may  be  written. 
P  =  total  load,  N  =  revs,  per  minute. 

Projected  area  =  7rJmX.i 5*4,.    . 

P 


.471^ 


*  See  Proc.  Inst.  M.   E.,  1888  and  1891,  for   reports   on    experiments  with 
thrust-bearings. 


JOURNALS,  BEARINGS,  AND   LUBRICATION.  241 

P2 


If  there  are  n  collars, 


i. 45  A 


For  pivot  bearings  of  the  general  type  shown  in  Fig.  128, 
Reuleaux  (Constructor,  p.  65)  gives  for  steel  on  bronze: 

Slow-moving  pivots,  ^=0.035  VP; 

Up  to  150  r.p.m.,     J  =  0.050 vjP; 

Above  150  r.p.m.,      d=o.oo4^PN. 

Tower's  experiments  on  pivot  bearings  of  this  form  under 
conditions  of  continuous  lubrication  show  that  the  best  result? 
are  attained  when 

,  V  being  the  outer  circumferential  velocity. 

Let  d  =  diameter  of  pivot  sought  for  a  load  =  P,  and  a  speed 
=  N,  r.p.m. 

P 


d= 


242 


MACHINE  DESIGN. 


If  this  relationship  of  p,  V,  d,  and  N  obtains  and  the  ar- 
rangement provides  for  continuous  lubrication  as  shown,  even 
if  there  are  no  loose  rings  or  disks,  a  coefficient  of  friction,  /z, 
as  low  as  0.005  may  be  expected.  This  arrangement  gives 
film  lubrication,  but  not  in  its  most  perfect  form. 

The  following  table  may  be  used  as  an  approximate  guide 
in  the  designing  of  thrust-bearings.  The  material  of  the  thrust- 
journal  is  wrought  iron  or  steel,  and  the  bearing  is  of  bronze 
or  brass  (babbitt  metal  is  seldom  used  for  this  purpose). 

TABLE  XVII.— THRUST  BEARINGS 


Kind  of  Bearing 

Velocity  of  Rubbing  Surface 
ft.  per  min. 

Allowable   pressure   Ibs. 
per  sq.  in.  of  net  area 
(less  oil-grooves,  etc.) 

Solid  collar 

ioo-upward 

50-60 

Flat  Pivot,  bath  lubrication 

Slow  and  intermittent 

1000-1500 

Flat  Pivot,  bath  lubrication 

100 

I00*-6oo 

Flat  Pivot,  bath  lubrication 

200 

140* 

Flat  Pivot,  bath  lubrication 

400 

200* 

Flat  Pivot,  bath  lubrication 

800 

280* 

Flat  Pivot,  bath  lubrication 

1600 

400* 

Loose  Ring,  drill  spindle,  or 

worm  shaft 

336 

*  For  best  efficiency. 

If  the  journal  is  of  cast  iron  and  runs  on  bronze  or  brass,  the 
values  of  allowable  pressure  given  should  be  divided  by  two. 

The  most  efficient  forms  of  thrust-bearings  are  those  *  employ- 
ing the  principles  shown  in  Fig.  128. 

Between  the  end  of  the  shaft  and  the  bottom  of  the  step  a 
series  of  accurately  finished  disks  are  introduced.  The  disks  are 
alternately  hard  steel  and  bronze,  the  top  one  is  fastened  to  the 
shaft,  the  lower  to  the  step,  and  the  rest  are  free.  As  indicated, 
each  disk  has  a  hole  through  the  middle  and  radial  grooves  to 
permit  the  lubricant  to  have  access  between  the  disks.  The 


*  See  Trans.  A.  S.  M.  E.,  Vol.  VI,  p.  852,  and  Proc.  Inst.  M.  E.,    1888,  p. 
184;    1891,  Plate  30. 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


243 


effect  of  centrifugal  force  when  the  shaft  is  rotating  is  to  force 
the  oil  outward  from  between  the  plates  and  upward.  It  is 
collected  in  the  annular  chamber  a-a  and  flows  from  there 
down  the  drilled  passages  back  to  the  bottom  of  the  bearing. 
This  is  equivalent  to  a  continuous  automatic  pump  action  sup- 
plying oil  to  the  surfaces.  This  form  of  bearing  reduces  the 
relative  motion  between  successive  surfaces  to  a  minimum.  A 
similar  arrangement  of  loose  disks  can  be  used  to  great  advan- 
tage on  small  propeller  shafts  and  on  worm  shafts. 

For  thrust-bearings  in  which  the  lubricant  is  automatically 
circulated,  or  supplied  by  a  force-pump  so  as  to  "  float "  the 


FIG.  128. 


journal,  the  allowable  unit  bearing  pressures  become  quite  great, 
examples  of  satisfactory  operation  at  loads  as  high  as  1000  Ibs. 
per  sq.in.  being  known.  With  a  lubricant  of  suitable  viscosity 
the  conditions,  at  sufficiently  high  speeds,  would  tend  to  give 
practically  fluid  friction,  i.e.,  the  frictional  resistance  would  be 
independent  of  the  pressure.  As  the  speed  decreases  the  tend- 
ency to  maintain  the  oil- film  grows  less,  however,  and  there  are 
critical  speeds  corresponding  to  certain  loads  at  which  the  film 
appears  to  break  down  and  seizing  takes  place.  For  pivot 
bearings  this  minimum  speed  appears  to  be,  from  Tower's  experi- 


244 


MACHINE  DESIGN. 


P2 

ments,   V  =  ~.     Where  special  means  of  forced  lubrication  are 

not  employed  it  will  be  safe  in  the  design  of  ordinary  thrust-jour- 
nals to  use  the  unit  pressures  given  in  Table  XVII. 

Fig.    i2&4    shows   the   step   bearing   of   the    Curtis   turbine 


FIG.  1 2  84. 


employing  forced  lubrication.     The  following  data  *  will  be  found 
of  interest  in  connection  with  the  design  of  this  type  of  bearing: 


Rating. 

Gals,  per  min. 

Unbalanced 
weight. 

Step  block. 

K.W. 

R.p.m. 

Total. 

i.e.  load. 

Outside 

Inside 

diameter. 

diameter. 

750 

1800 

7 

10800 

8 

5l 

1500 

1500 

8 

19700 

ii 

8 

3750 

900 

16 

65000 

16 

9i 

5000 

750 

21 

IOIOOO 

16 

10 

9000 

750 

27 

148000 

183 

i  of 

14000 

750 

32 

190000 

20 

13^ 

15000 

750 

36 

216000 

21 

IS 

2OOOO 

75° 

43 

233000 

21 

i5 

From  Alford's  Bearings,  McGraw-Hill  Co. 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  245 

When  bearings  have  to  be  used  where  corrosion  or  electro- 
lytic action  is  to  be  feared,  as  in  turbine  work,  glass  and  the 
end  grain  of  very  hard  woods  have  been  used  successfully  as 
bearing  materials. 

140.  Problem.  —  It  is  required  to  design  the  collar  thrust- 
journal  that  is  to  receive  the  propelling  pressure  from  the  screw 
of  a  small  yacht.  The  necessary  data  are  as  follows:  The 
maximum  power  delivered  to  the  shaft  is  70  H.P.;  pitch  of 
screw  is  4  feet;  slip  of  screw  is  20  per  cent;  shaft  revolves  250 
times  per  minute;  diameter  of  shaft  is  4  inches. 

For  every  revolution  of  the  screw  the  yacht  moves  forward 
a  distance  =  4  feet  less  20  per  cent  «=  3.  2  feet,  and  the  speed  of 
the  yacht  in  feet  per  minute  =  2  50X3.  2  =800. 

70  H.  P.  =  70X33,000  =  2,310,000  ft.-lbs.  per  minute. 

This  work  may  be  resolved  into  its  factors  of  force  and  space, 
and  the  propelling  force  is  equal  to  2,310,000-^-800  =  2900  Ibs., 
nearly. 

The  shaft  is  4  inches  in  diameter,  and  the  collars  must  project 
beyond  its  surface.  Estimate  that  the  mean  diameter  of  the 
rubbing  surface  is  4.5  inches,  then  the  mean  velocity  of  rubbing 

surface  would  equal  4.5  X  —  X  250  =  294   feet   per  minute.     A 

12 

safe  value  of  p,  the  pressure  per  square  inch,  at  this  speed  is 
50  Ibs.  The  necessary  area  of  the  journal  surface  is  there- 
fore =  2900-7-  50  =  58  square  inches. 

It  has  been  seen  that  it  is  desirable  to  keep  the  radial  dimen- 
sion of  the  collar  surface  as  small  as  possible  in  order  to  have 
as  nearly  the  same  velocity  at  all  parts  of  the  rubbing  surface 
as  possible.  The  width  of  collar  in  this  case  will  be  assumed 
=^^=0.75  inch;  then  the  bearing  surface  in  each  collar 


=  23.7  -12.5  =  n.2sq.m. 

Then   the    number   of    collars    equals    the    total    required    area 
divided  by  the  area  of  each  collar  =58-=-  11.2=  5.18,  say  6. 


246 


MACHINE  DESIGN. 


141.   Thrust-bearings    with    Perfect    Film    Lubrication. — In 

recent  years  a  great  advance  has    been  made  in  the  design  of 
thrust-bearings  by  succeeding  in  applying  to  them  the  principles 


Position  of  Shoe  in  Bearing. 
FIG.  I2&B. 

of  continuous  perfect  film  lubrication.  The  mathematical  analysis 
was  first  published  by  Mitchell,*  but  the  earliest  practical  bear- 

*  Zeit.  fur  Math,  und  Physik,  1905. 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  247 

ings  designed  for  use  appear  to  be  those  of  Kingsbury  *  and 
this  type  of  bearing  is  known  by  the  latter's  name.  The  bearing 
is  submerged  in  oil.  The  ring  which  supports  the  step  or  collar 
is  not  one  solid  ring,  but  is  divided  into  segments  each  one  of 
which  is  pivoted  at  or  near  its  center  of  pressure  on  a  spherical- 
ended,  cylindrical,  upright  support.  As  a  consequence,  each 
individual  segment  is  free  to  incline  at  a  small  angle  (about 
1/3000)  with  the  step  surface,  allowing  the  perfect  formation 
of  the  lubricant  wedge. 

Steam  turbines  with  these  bearings  have  been  run  with  a 
unit  pressure  of  500  Ibs.  per  square  inch  at  linear  speeds  of  3000 
to  4500  ft.  per  minute,  and  vertical  water  wheels  with  a  unit 
pressure  of  250  to  400  Ibs.  per  square  inch  with  a  coefficient 
of  friction  generally  lying  between  the  remarkably  low  values 
of  .001  and  .002.  This  equals  or  exceeds  ball  or  roller  bearing 
efficiency.  Tests  show  that  there  is  practically  no  limit  of  speed 
provided  that  the  oil  be  circulated  and  the  heat  generated  in  it 
by  friction  be  removed.  It  will  probably  be  found  that  the 
allowable  pressure  will  lie  in  the  vicinity  of  p  =  2oV  V.  These 
bearings  have  shown  great  overload  capacity  under  test. 

The  center  of  pressure,  for  square  blocks  or  those  whose 
length  is  not  more  than  3  times  their  width,  is  about  .4  of  their 
length  from  the  rear  end.f  This  is  the  proper  location  of  the 
spherical  seat  of  each  segment. 

142.  Bearings  and  Boxes. — The  function  of  a  bearing  or  box 
is  to  insure  that  the  journal  with  which  it  engages  shall  have 
an  accurate  motion  of  rotation  or  vibration  about  the  given  axis. 
It  must  therefore  fit  the  journal  without  lost  motion;  must 
afford  means  of  taking  up  the  lost  motion  that  results  neces- 
sarily from  wear  ;  must  resist  the  forces  that  come  upon  it 
through  the  journal,  without  undue  yielding  ;  must  have  the 

*See  Alford's  "Bearings,"  p.  150,  and  Engineering  Record,  Jan.  n  and  18, 
t  Zeit.  fur  Math,  und  Physik,  1905. 


248  MACHINE  DESIGN. 

wearing  surface  of  such  material  as  will  run  in  contact  with 
the  material  of  the  journal  with  the  least  possible  friction  and 
least  tendency  to  heating  and  abrasion;  and  must  usually 
include  some  device  for  the  maintenance  of  the  lubrication.  The 
selection  of  the  materials  and  the  providing  of  sufficient  strength 
and  stiffness  depends  upon  principles  already  considered,  and 
so  it  remains  to  discuss  the  means  for  the  taking  up  of  necessary 
wear  and  for  providing  lubrication. 

Boxes  are  sometimes  made  solid  rings  or  shells,  the  journal 
being  inserted  endwise.  In  this  case  the  wear  can  only  be 
taken  up  by  making  the  engaging  surfaces  of  the  box  and  journal 
conical,  and  providing  for  endwise  adjustment  either  of  the 
box  itself  or  of  the  part  carrying  the  journal.  Thus,  in  Fig.  129, 

the  collars  for  the  pre- 
venting of  end  motion 
while  running  are  jam- 
nuts,  and  looseness  be- 
tween the  journal  and 
FIG.  129.  FIG.  130.  box  may  be  taken  up  by 

moving  the  journal  axially  toward  the  left. 

By  far  the  greater  number  of  boxes,  however,  are  made  in 
sections  and  the  lost  motion  is  taken  up  by  moving  one  or  more 
sections  toward  the  axis  of  rotation.  The  tendency  to  wear  is 
usually  in  one  direction,  and  it  is  sufficient  to  divide  the  box 
into  halves.  Thus,  in  Fig.  130,  the  journal  rotates  about  the 
axis  O,  and  all  the  wear  is  due  to  the  pressure  P  acting  in  the 
direction  shown.  The  wear  will  therefore  be  at  the  bottom  of 
the  box.  It  will  suffice  for  the  taking  up  of  wear  to  dress  off 
the  surfaces  at  aa,  and  thus  the  box-cap  may  be  drawn  further 
down  by  the  bolts,  and  the  lost  motion  is  reduced  to  an  admis- 
sible value.  " Liners, "  or  "shims,"  which  are  thin  pieces  of 
sheet  metal,  may  be  inserted  between  the  surfaces  of  division 
of  the  box  at  aa,  and  may  be  removed  successively  for  the  lower- 


JOURNALS,  BEARINGS,  AND   LUBRICATION. 


249 


ing  of  the  box-cap  as  the  wear  renders  it  necessary.  If  the  axis 
of  the  journal  must  be  kept  in  a  constant  position,  the  lower 
half  of  the  box  must  be  capable  of  being  raised. 

Sometimes,  as  in  the  case  of  the  box  for  the  main  journal  of 
a  steam-engine  shaft,  the  direction  of  wear  is  not  constant. 
Thus,  in  Fig.  131,  A  represents  the  main  shaft  of  an  engine. 
There  is  a  tendency  to  wear  in  the  direction  B, 
because  of  the  weight  of  the  shaft  and  its  at- 
tached parts;  there  is  also  a  tendency  to  wear 
because  of  the  pressure  that  comes  through  the 
connecting-rod  and  crank.  The  direction  of 
this  pressure  is  continually  varying,  but  the 
average  directions  on  forward  and  return  stroke 
may  be  represented  by  C  and  D.  Provision  needs  to  be  made, 
therefore,  for  the  taking  up  of  wear  in  these  two  directions.  If 


FIG.  132. 

the  box  be  divided  on  the  line  EF,  wear  will  be  taken  up  verti- 
cally and  horizontally  by  reducing  the  liners.  Usually,  however, 
in  the  larger  engines  the  box  is  divided  into  four  sections,  A}  B, 


250 


MACHINE  DESIGN. 


C,  and  D  (Fig.  132),  and  A  and  C  are  capable  of  being  moved 
toward  the  shaft  by  means  of  screws  or  wedges,  while  D  may  be 
raised  by  means  of  the  insertion  of  "  shims. ' ' 

The  lost  motion  between  a  journal  and  its  box  is  sometimes 
taken  up  by  making  the  box  as  shown  in  Fig.  133.  The  exter- 
nal surface  of  the  box  is  conical  and  fits  in  a  conical  hole  in 
the  machine  frame.  The  box  is  split  entirely  through  at  A, 
parallel  to  the  axis,  and  partly  through  at  B  and  C.  The  ends 
of  the  box  are  threaded,  and  the  nuts  E  and  F  are  screwed  on. 
After  the  journal  has  run  long  enough  so  that  there  is  an  unal- 
lowable amount  of  lost  motion,  the  nut  F  is  loosened  and  E 
is  screwed  up,  the  effect  being  to  draw  the  conical  box 
further  into  the  conical  hole  in  the  machine  frame;  the  hole 


FIG.  133. 

through  the  box  is  thereby  closed  up  and  lost  motion  is  reduced. 
After  this  operation  the  hole  cannot  be  truly  cylindrical,  and 
if  the  cylindrical  form  of  the  journal  has  been  maintained,  it 
will  not  have  a  bearing  throughout  its  entire  surface.  This  is 
not  usually  of  very  great  importance,  however,  and  the  form  of 
box  has  the  advantage  that  it  holds  the  axis  of  the  journal  in 
a  constant  position.  As  far  as  is  possible  the  box  should  be 
so  designed  as  to  exclude  all  dust  and  grit  from  the  bearing 
surfaces. 

All  boxes  in  self-contained  machines,  like  engines  or  machine 
tools,  need  to  be  rigidly  supported  to  prevent  the  localization 
of  pressure,  since  the  parts  that  carry  the  journals  are  made  as 
rigid  as  possible.  In  line  shafts  and  other  parts  carrying  journals, 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  251 

when  the  length  is  great  in  comparison  to  the  lateral  dimensions, 
some  yielding  must  necessarily  occur,  and  if  the  boxes  were 
rigid,  localization  of  pressure  would  result.  Hence  "self- 
adjusting"  boxes  are  used.  A  point  in  the  axis  of  rotation  at 
the  center  of  the  length  of  the  box  is  held  immovable,  but  the 
box  is  free  to  move  in  any  way  about  this  point,  and  thus  adjusts 
itself  to  any  yielding  of  the  shaft.  This  result  is  attained  as 
shown  in  Fig.  134.  O  is  the  center  of  the  motion  of  the  box; 


FIG.  134. 

B  and  A  are  spherical  surfaces  formed  on  the  box,  their  center 
being  at  O.  The  support  for  the  box  contains  internal  spherical 
surfaces  which  engage  with  A  and  B.  Thus  the  point  O  is  always 
held  in  a  constant  position,  but  the  box  itself  is  free  to  move  in 
any  way  about  O  as  a  center.  Therefore  the  box  adjusts 
itself  within  limits  to  any  position  of  the  shaft  and  hence  the 
localization  of  pressure  is  impossible. 

In  thrust-bearings  for  vertical  shafts  the  weight  of  the  shaft 
and  its  attached  parts  serves  to  hold  the  rubbing  surfaces  in 


252  MACHINE  DESIGN. 

contact  and  the  lost  motion  is  taken  up  by  the  shaft  following 
.down  as  wear  occurs.  In  collar  thrust-bearings  for  horizontal 
shafts  the  design  is  such  that  the  bearing  for  each  collar  is 
separate  and  adjustable.  The  pressure  on  the  different  collars 
may  thus  be  equalized.* 

143.  Lubrication  of  Journals,  f — The  best  method  of  lubrica- 
tion is  that  in  which  the  rubbing  surfaces  are  constantly  sub- 
merged in  a  bath  of  lubricating  fluid.  This  method  should  be 
employed  wherever  possible  if  the  pressure  and  surface  velocity 
are  high.  Unfortunately  it  cannot  be  used  in  the  majority  of 
cases.  It  is  not  necessary  that  the  whole  surface  be  sub- 
merged. If  a  part  of  the  moving  surface  runs  in  the  oil-bath  it 
is  sumcient.t  The  same  result  is  accomplished  by  the  use  of 
chains  and  rings  encircling  the  journals  and  dipping  into  oil- 
pockets,  as  described  later  in  this  section.  The  effect  is  to 
form  a  complete  film  of  oil  enveloping  the  journal.  To  allow 
this  it  is  evident  that  the  bore  of  the  bearing  must  be  slightly 
greater  than  the  diameter  of  the  journal  and  a  good  value  to  use 
for  "  running  fit  allowances  "  is  o.ooi  inch  per  inch  of  diameter. 
The  oil  film  may  be  conceived  to  be  made  up  of  a  series  of 
layers,  the  one  next  the  bearing  surface  remaining  stationary 
with  regard  to  it,  while  the  layer  in  immediate  contact  with  the 
shaft  rotates  with  the  latter.  The  intermediate  layers,  therefore, 
slip  upon  each  other  as  the  shaft  rotates  and  the  friction  becomes 
very  closely  akin  to  "  fluid  friction"  with  the  bearing  floating 

*  For  complete  and  varied  details  of  marine  thrust-bearings  see  '  Maw's 
Modern  Practice  in  Marine  Engineering." 

t  See  "  Lubrication  and  Lubricants,"  by  Archbutt  and  Deeley,  London. 

|  Tower's  experiments,  Proc.  Inst.  M.  E.,  1883  and  1885.  See  further  Prof. 
Reynolds'  paper  "On  the  Theory  of  Lubrication,"  Phil.  Trans.,  1886,  Part  I, 
pp.  157-234- 

§  Professor  Reynolds  states,  in  Phil.  Trans.,  1886,  Part  i,  p.  161,  that  if  viscosity 
were  constant  the  friction  would  be  inversely  proportional  to  the  difference  in  radii 
of  the  journal  and  the  bearing. 


254 


MACHINE  DESIGN. 


upon  the  lubricant,  there  being  no  contact  between  the  metallic 
surfaces.  Fig.  134  A  shows  the  conditions  of  pressure  existing 
in  the  film  in  Tower's  classic  experiments.  It  is  impossible 
to  introduce  oil  satisfactorily  at  the  points  where  the  film  is 
under  pressure;  it  should  be  introduced  and  distributed  where 


TOP  UA.L.F  FlG.  1346.  BOTTOM  HALF 

the  pressure  is  least.  Referring  back  to  Fig.  122^,  d,  it  will 
be  seen  that  this  point  is  at  C,  just  beyond  B,  the  point  of  nearest 
approach.  Dewrance  (Proc.  Inst.  Civil  Engs.,  1896)  reports  as 
high  as  30  inches  vacuum  at  this  point  on  a  heavily  loaded  journal. 
Other  experimenters  confirm  this  phenomenon. 

Under  the  action  of  the  load  the  edges  of  the  boxes  tend 
to  "  pinch  in  "  and  scrape  off  the  film  from  the  journal.  To 
prevent  this  these  edges  should  be  cut  away,  thus  also  forming 
an  excellent  oil  channel  for  longitudinal  distribution  of  the  oil 
where  the  pressure  is  least.  An  excellent  arrangement  of  boxes 
for  distributing  the  oil  and  maintaining  the  film  is  shown  in  Fig. 
134$,  which  is  copied  from  Vol.  27,  Trans.  A.  S.  M.  E.,  p.  484. 

With  pad  lubrication  or  where  the  oil  is  fed  drop  by  drop  there 
is  a  tendency  for  the  film  to  be  too  thin  or  to  break  down,  allow- 


JOURNALS,  BEARINGS,  AND  LUBRICATION.  255 

ing  contact  of  the  metallic  surfaces,  and  the  highly  favorable 
condition  of  fluid  friction  disappears.  The  conditions  then  lie 
between  "  fluid  friction  "  and  "  solid  friction  "  and  are  too  com- 
plex for  the  statement  of  consistent  results,  but  it  may  be  approxi- 
mately stated  that,  with  good  pad  lubrication,  the  coefficient 
of  friction  will  be  about  twice  that  of  film  lubrication.  With 
drop  by  drop  lubrication  the  value  of  the  coefficient  may  be- 
come anything  between  twice  that  for  best  film  lubrication 
(i.e.,  =  0.0012),  and  0.18,  the  value  determined  by  Morin  for 
dry  journals.  It  becomes  apparent  that  some  system  of  forced 
or  flooded  lubrication  whereby 

a  continuous   film  is  insured  is  A B c 

of  utmost  value  in  maintaining 
efficiency. 

Let  /,  Fig.  135,  represent  a    | 
journal  with  its  box,  and  let  A, 
B,  and  C  be  oil-holes.     If  oil  is  FIG.  135. 

introduced  into  the  hole  A,    it 

will  tend  to  flow  out  from  between  the  rubbing  surfaces  by  the 
shortest  way,  i.e.,  it  will  come  out  at  D.  A  small  amount 
will  probably  go  toward  the  other  end  of  the  box  because  of 
capillary  attraction,  but  usually  none  of  it  will  reach  the  middle 
of  the  box.  If  oil  be  introduced  at  C,  it  will  come  out  at  E.  A 
constant  feed,  therefore,  might  be  maintained  at  A  and  C,  and 
yet  the  middle  of  the  box  might  run  dry.  If  the  oil  be  introduced 
at  B,  however,  it  tends  to  flow  equally  in  both  directions,  and 
the  entire  journal  is  lubricated.  The  conclusion  follows  that 
oil  ought,  when  possible,  to  be  introduced  at  the  middle  of  the 
length  of  a  cylindrical  journal.  It  should  be  introduced  as  far 
as  possible  from  the  side  where  the  forces  press  the  journal  and 
box  closest  together.*  If  a  conical  journal  runs  at  a  high  velocity, 
the  oil  under  the  influence  of  centrifugal  force  tends  to  go  to 

*  Tower's  experiments,  Proc.  Inst.  M.  E.,  1883  and  1885. 


256 


MACHINE  DESIGN 


the  large  end  of  the  cone,  and  therefore  the  oil  should  be  intro- 
duced at  the  small  end  to  insure  its  distribution  over  the  entire 
journal  surface. 

If  the  end  of  a  vertical  thrust-journal  whose  outline  is  a 
cone  or  tractrix,  as  in  Fig.  136,  dips  into  a  bath  of  oil,  B,  the 
oil  will  be  carried  by  its  centrifugal  force,  if  the  velocity  be 
high,  up  between  the  rubbing  surfaces,  and  will  be  delivered 
into  the  groove  A  A.  If  holes  connect  A  and  B,  gravity  will 
return  the  oil  to  B,  and  a  constant  circulation  will  be  main- 
tained. If  the  thrust-journal  has  simply  a  flat  end,  as  in  Fig. 
137,  the  oil  should  be  supplied  at  the  center  of  the  bearing; 
centrifugal  force  will  then  distribute  it  over  the  entire  surface. 
If  the  oil  is  forced  in  under  a  pressure  sufficient  to  "float"  the 
shaft  the  friction  will  be  greatly  reduced.  Vertical  shaft  thrust- 
journals  may  usually  be  arranged  to  run  in  an  oil-bath.  Marine 
collar  thrust-journals  are  always  arranged  to  run  in  an  oil-bath. 


FIG.  136. 


FIG  137. 


Z/y///A 


FIG.  138. 


Sometimes  a  journal  is  stationary,  and  the  box  rotates 
about  it,  as  in  the  case  of  a  loose  pulley,  Fig.  138.  If  the  oil 
is  introduced  into  a  tube,  as  is  often  done,  its  centrifugal 
force  will  carry  it  away  from  the  rubbing  surface  unless  a  "  grease 
candle  "  or  other  type  of  pressure  lubricating  device  is  used.  But 
if  a  hole  is  drilled  in  the  axis  of  the  journal,  the  lubricant  intro- 


JOURNALS,  BEARINGS,  AND  LUBRICATION. 


257 


duced  into  it  will  be  carried  to  the  rubbing  surfaces  as  required. 
If  a  journal  is  carried  in  a  rotating  part  at  a  considerable  dis- 
tance from  the  axis  of  rotation,  and  it  requires  to  be  oiled  while 
in  motion,  a  channel  may  be  provided  from  the  axis  of  rota- 
tion, where  oil  may  be  introduced  conveniently,  to  the  rub- 
bing surfaces,  and  the  oil  will  be  carried  out  by  centrifugal 
force.  Thus  Fig.  139  shows  an  engine-crank  in  section.  Oil 
is  introduced  at  b,  and  centrifugal  force  carries  it  through 
the  channel  provided  to  a,  where  it  serves  to  lubricate  the  rub- 


FIG.  139. 


FIG.  140. 


bing  surfaces  of  the  crank-pin  and  its  box.  If  a  journal  is 
carried  in  a  reciprocating  machine  part,  and  requires  to  be 
oiled  while  in  motion,  the  4 'wick -and  wiper  "  method  is  one  of 
the  best.  (See  Fig.  140.)  An  ordinary  oil-cup  with  an  adjust- 
able feed  is  mounted  in  a  proper  position  opposite  the  end  of 
the  stroke  of  the  reciprocating  part,  and  a  piece  of  flat  wick 
projects  from  its  delivery-tube.  A  drop  of  oil  runs  down  and 
hangs  suspended  at  its  end.  Another  oil  cup  is  attached  to 
the  reciprocating  part,  which  carries  a  hooked  "  wiper,"  C. 
The  delivery-tube  from  C  leads  to  the  rubbing  surfaces  to  be 
lubricated.  When  the  reciprocating  part  reaches  the  end  of 
its  stroke  the  wiper  picks  ofT  the  drop  of  oil  from  the  wick 


258  MACHINE  DESIGN. 

and  it  runs  down  into  the  oil-cup  C,  and  thence  to  the  sur- 
faces to  be  lubricated.  This  method  applies  to  the  oiling  of 
the  cross-head  pin  of  a  steam-engine.  The  same  method  is 
sometimes  applied  to  the  crank-pin,  but  here,  through  a  part 
of  the  revolution,  the  tendency  of  the  centrifugal  force  is  to 
force  the  oil  out  of  the  cup,  and  therefore  the  plan  of  oiling 
from  the  axis  is  probably  preferable. 

When  journals  are  lubricated  by  feed-oilers,  and  are  so 
located  as  not  to  attract  attention  if  the  lubrication  should  fail 
for  any  reason,  "  tallow-boxes  "  or  "  grease-cups  "  are  used. 
These  are  cup-like  depressions  usually  cast  in  the  box-cap 
and  communicating  by  means  of  an  oil-hole  with  the  rubbing 
surface.  These  cups  are  filled  with  grease  that  is  solid  at 
the  ordinary  temperature  of  the  box,  but  if  there  is  the  least 
rise  in  temperature  because  of  the  failure  of  the  oil- supply, 
the  grease  melts  and  runs  to  the  rubbing  surfaces,  and  sup- 
plies the  lubrication  temporarily.  This  safety  device  is  used 
very  commonly  on  line-shaft  journals. 

The  most  common  forms  of  feed-oilers  are:  I.  The  oil-cup 
with  an  adjustable  valve  that  controls  the  rate  of  flow.  II.  The 
oil-cup  with  a  wick  feed  (Fig.  141).  The  delivery  has  a  tube 
inserted  in  it  which  projects  nearly  to  the  top  of  the  cup.  In 
this  tube  a  piece  of  wicking  is  inserted,  and  its  end  dips  into 
the  oil  in  the  cup.  The  wick,  by  capillary  attraction,  carries 
the  oil  slowly  and  continuously  over  through  the  tube  to  the 
rubbing  surfaces.  III.  The  cup  with  a  copper  rod  (Fig.  142). 
The  oil-cup  is  filled  with  grease  that  melts  with  a  very  slight 
elevation  of  temperature,  and  A  is  a  small  copper  rod  dropped 
into  the  delivery-tube  and  resting  on  the  surface  of  the  journal. 
The  slight  friction  between  the  rod  and  the  journal  warms 
the  rod  and  it  melts  the  grease  in  contact  with  it,  which  runs 
down  the  rod  to  the  rubbing  surface.  IV.  Sometimes  a  part 
of  the  surface  of  the  bottom  half  of  the  box  is  cut  away  and 


JOURNALS,  RE  A  RINGS,  AND  LUBRICATION. 


259 


a  felt  pad  is  inserted,  its  bottom  being  in  contact  with  an  oil- 
bath.  This  pad  rubs  against  the  surface  of  the  journal,  is 
kept  constantly  soaked  with  oil,  and  maintains  lubrication. 

Ring-and-chain  lubrication  may  be  considered  as  special 
forms  of  bath  lubrication.  Fig.  143  shows  a  ring  oiling  bearing. 

A  loose  ring  rests  on  top  of  the  journal,  the  upper  box  being 
cut  away  to  permit  this;  the  ring  surrounds  the  lower  box 


FIG.  141. 


FIG.  142. 


FIG.  143. 


and  extends  into  a  reservoir  filled  with  oil.  The  rotation  of 
the  shaft  carries  the  ring  with  it,  which,  in  turn,  brings  up  a 
constant  supply  of  oil  from  the  reservoir.  The  annular  spaces 
A- A  catch  all  oil  which  works  out  along  the  shaft  and  return 
it  to  the  reservoir. 

Modern  machines  are  equipped  frequently  with  complete 
oil  distributing  and  circulating  systems,  including  necessary  oil- 
pipes,  chambers,  cooling  and  filtering  devices  and  pump.  Con- 
tinuous lubrication  of  this  sort,  properly  applied,  leads  to  very 
high  mechanical  efficiencies. 

Graphite  is  winning  a  deservedly  high  place  as  a  lubricant  for 
certain  conditions.  Its  action  is  to  reduce  '•  solid  friction"'  by 


260  MACHINE  DESIGN. 

filling  the  inequalities  in  the  surfaces  of  the  relatively  moving 
members,  giving  each  a  smooth,  slippery  coating,  thereby  reduc- 
ing the  coefficient  of  friction.  It  is  particularly  useful  when  the 
conditions  of  pressure  or  temperature  are  such  as  would  tend 
to  squeeze  out,  gum,  or  destroy  liquid  lubricants,  if  these  were 
used  alone. 

Although  it  may  be  applied  in  some  cases  in  dry  flake  form, 
it  is  customary  to  use  it  in  the  form  of  a  mixture  with  oils,  grease, 
or  even  water.  Caution  must  be  observed  that  the  graphite  used 
is  free  from  all  grit. 


CHAPTER  XIII. 


ROLLER-  AND   BALL-BEARINGS. 

144.  General  Considerations. — By  substituting  rolling  motion 
in  bearings  in  place  of  relative  sliding,   friction  losses  can  be 
greatly    reduced.     In    the    design    of    such    bearings    there    are 
five  points  to  be  borne  in  mind: 

I.  The   arrangement  of  the   parts   and  their  form  must  be 
such  that  their  relative  motion  is  true  rolling  with  the  least  pos- 
sible amount  of  sliding.     This  means  spheric  motion. 

II.  The  form  of  the  constraining  surfaces  must  be  such  that 
the  rolling  parts  will  not  have  any  effective  tendency  to  leave 
the  proper  guides  or  "  races." 

III.  The  rollers  and  balls  must  not  be  unduly  loaded. 

IV.  Provision  must  be  made  to  admit  the  lubricant,  and  to 
exclude  all  dust  and  grit. 

V.  The  arrangement  of  the  parts  must  be  such  as  will  permit 
unavoidable  elastic  yielding  without  causing  pinching  or  binding. 

These  points  will  bs  considered  in  the  order  given. 

145.  I.  Rolling,  Sliding,  and  Spinning.     (See  Fig.  144.) — At 


A  is  shown  the  longitudinal  section  of  a  cylindrical  ball-bearing 
of  the  simplest   form  stripped   of  all  auxiliary   parts.      At  B 

261 


262 


MACHINE  DESIGN. 


is  shown  the  same  for  a  roller-bearing.  At  C  is  a  cross-section  of 
either,  showing  but  one  pair  of  balls  or  rollers,  R.  S  is  the  journal 
and  T  the  box.  Consider  T  as  stationary,  then  the  point  of 
contact  of  R  and  S  would  have  the'  same  motion  relative  to  T 
whether  considered  as  a  point  of  R  or  of  5,  and  if  the  surface 
friction  were  sufficient  there  would  be  no  reason  for  slippage. 
As  a  matter  of  fact,  in  the  actual  bearing  there  will  be  a  slight 
amount  of  slipping  .at  both  of  the  points  of  contact.  This  form 
of  bearing  is  called  the  "  two-point  bearing,"  because  there  are 
two  points  of  contact.  All  cylindrical  roller-bearings  are  of 
this  fundamental  form.  In  order  to  have  them  of  practical  use 
the  rollers  must  be  held  in  a  case  or  "cage"  so  that  their  axes 
will  always  remain  parallel  with  the  axis  of  the  shaft.  Fig.  145 
shows  such  a  "cage"  with  rollers  in  place. 


FIG.  145. 

Since  the  rollers  are  generally  of  hardened  and  ground  steel 
the  best  service  with  the  least  wear  wrill  be  given  when  the 
engaging  surfaces  are  of  the  same  material.  To  meet  this  when 
the  shaft  is  of  soft  steel,  say,  and  the  box  of  cast  iron,  a  hardened 
and  ground-steel  ring  is  fitted  over  the  shaft  as  a  shell  and 
another  inside  the  box  as  a  bushing,  and  the  rollers  run  between 
the  outer  surface  of  the  former  and  the  inner  surface  of  the 
latter.  The  necessary  room  for  "  play  "  makes  it  inevitable  that 
the  axes  of  roller  bearings  will  get  out  of  line.  If  the  rollers  are 
long  enough  the  resultant  forces  will  break  them  unless  they  are 


ROLLER-  AND  BALL-BEARINGS. 


263 


flexible,  i.e.,  made  of  helically  rolled  strips.  The  alternative 
expedient  is  that,  shown  in  Fig.  145,  of  dividing  the  rollers  into 
short,  separate  lengths. 

Ball-bearings  are  subject  to  an  action  known  as  "  spinning." 
To  illustrate  this,  consider  the  three-point  ball-bearing  shown 
in  Fig.  146.  Here  the  centres  are  as  shown  in  B,  and  the  con- 
ditions are  correct  for  theoretical  rolling  as  long  as  point  contact 
is  maintained  and  axis  C-D  remains  parallel  to  axis  E-F.  But 
when  the  bearing  is  in  use  the  points  of  contact,  on  each  side 
of  R,  with  T  become  small  areas,  as  shown  in  B.  Considering 
the  relative  motion  of  R  and  T  at  any  instant  it  will  be  seen  that 


FIG.  146. 


FIG.  147. 


there  is  an  action  on  each  side  of  the  ball  akin  to  that  of  a  small 
thrust-bearing.  The  rubbing  produced  in  this  manner  naturally 
causes  undesirable  friction.  This  is  the  action  known  as  "spin- 
ning." It  may  also  be  called  boring.  Spinning  or  boring  may 
be  denned  as  that  action  of  the  ball  with  relation  to  the  con- 
straining surfaces  which  results  from  a  rotation  of  the  ball  about 
an  instantaneous  axis  that  is  approximately  normal  (actually, 
anything  but  tangent)  to  the  constraining  surfaces. 

Obviously  it  is  even  more  marked  in  the  case  of  a  four- 
point  bearing,  as  shown  in  Fig.  147. 

Here,  also,  there  is  pure  rolling  motion  as  long  as  point  con- 
tact is  maintained,  and  the  axes  C-D  and  E-F  remain  parallel 
to  axis  G-H\  but  as  soon  as  the  load  is  applied  the  points  of 


264 


MACHINE   DESIGN. 


contact  become  areas,  and  "  spinning "  results  at  four  surfaces. 
Experiments  bear  out  the  conclusion  that  a  properly  designed 
two-point  bearing  will  have  less  friction  than  a  three-point,  and 
a  three-point  will  have  less  than  a  four-point. 

In  a  "  race  "  whose  radius  of  curvature  is  just  equal  to  that 
of  the  ball  the  friction  becomes  excessive.  Such  races  should 
never  be  used.  (See  Fig.  148.)  They  have  excessive  slippage. 

A  force  acting  at  the  surface  of  a  ball  will  tend  to  rotate 
it  about  an  axis  parallel  to  the  tangent  plane  in  which  the  actu- 
ating force  lies;  furthermore,  this  axis  will  be  at  a  right  angle 
with  the  direction  of  the  force.  This  is  true  because  it  is  merely 
a  special  application  of  the  general  law  that  a  force  applied  to 
a  body  will  tend  to  move  it  in  the  direction  of  action  of  the  force. 
If  other  forces,  or  the  form  of  the  constraining  surfaces,  prevent 
rotation  about  this  axis  and  cause  it  to  take  place  about  some 
other  compromise  axis,  "  sliding "  takes  place  to  some  extent 
and  the  efficiency  and  life  of  the  bearing  are  lowered. 


fp 


FIG.  148. 


FIG.  149. 


The  general  law  for  the  form  of  rolling  bearings  may  now  be 
stated  as  follows: 

For  true  rolling,  the  constraining  surfaces  of  the  journal 
and  box  (i.e.,  the  "races")  must  be  so  formed  that  the  axes  of 
relative  rotation  of  the  rollers  or  balls  with  races  and  cage  will 
all  intersect  the  main  axis  of  the  bearing  at  a  fixed  point  through- 
out the  complete  revolution  of  the  journal.  This  may  be  made 
clear  by  examples. 


ROLLER-  AND  BALL-BEARINGS. 


265 


Fig.  149  shows  a  ball  or  roller  R  held  between  two  similar 
plates  T  and  S.  The  upper  plate,  T,  presses  down  on  R  with 
a  force  P  which  is  transmitted  through  R  to  5. 

By  the  principles  of  so-called  "  rolling  friction,"  to  roll 
T  on  R  will  require  a  force  fP  (i.e.,  proportional  to  P)  to 
overcome  the  resistance.  The  motion  of  T  on  R  causes  R 
to  roll  on  S,  to  which  rolling  there  is  induced  a  resistance  also 
equal  to  /P,  but  in  the  opposite  direction  as  regards  R.  These 
two  forces  being  equal,  opposite,  and  applied  at  the  same  dis- 
tance from  the  center  of  R,  form  a  couple  whose  effect  would 
be  to  give  R  a  motion  of  rotation  about  an  axis  through  its 
center,  and  perpendicular  to  the  plane  in  which  they  both  lie. 

This  case  is  similar  to  those  shown  in  Fig.  144,  except  that 
in  the  cases  there  shown  5  and  T  are  not  plane  surfaces.  Each 
ball  in  case  A  and  each  roller  in  case  B  tends  to  rotate  about  an 
axis  (relatively  to  the  "  cage,''  not  shown)  as  indicated  by  the 
dotted  lines.  In  both  cases  the  individual  axes  all  intersect 
the  main  axis  of  the  journal  at  a  fixed  point,  namely,  at  infinity, 
throughout  the  revolution.  The  general  law  for  true  rolling 
is  therefore  fulfilled. 

In  the  cases  shown  in  Figs.  146  and  147,  obviously  the 
same  conditions  hold. 

Next  consider  the  thrust-bearings  shown  in  Fig.  150: 

'  T 


FIG.  150. 

Take  case  A  first.  5  is  the  moving  member,  T  the  sta- 
tionary member,  R  one  of  the  balls,  and  OF  is  the  axis  of 
rotation  of  5  relatively  to  T.  The  center  of  the  ball  is  at  any 


266  MACHINE  DESIGN. 

distance  r  from  the  axis  O  F,  and  its  points  of  contact  with  S 
and  T  are  termed  A  and  B  respectively.  Relatively  to  the 
inclosing  cage  (not  shown)  all  parts  of  the  ball  in  obedience 
to  the  acting  force  tend  to  rotate  about  the  axis  OX,  which 
always  cuts  OF  at  O.  Relative  to  5,  the  ball,  R,  rotates  about 
the  instantaneous  axis  OA ;  relative  to  T,  about  OB.  The  three 
instantaneous  axes  of  relative'  motion  intersect  the  main  axis 
at  O.  It  is  not  essential  that  the  angle  XOY  be  a  right  angle. 
Theoretically  the  conditions  for  true  rolling  are  fulfilled.  Practically 
there  will  be  boring  between  the  outer  end  of  the  ball  axis  and 
the  cage.  The  greater  the  load  along  OF,  and  the  greater  the 
angle  AOB,  the  more  serious  this  becomes. 

In  case  B,  as  5  rotates  relative  to  T1,  the  point  D  common 
to  R  and  5  will  have  a  linear  velocity  proportional  to  r2  and, 
similarly,  C's  linear  velocity  will  be  proportional  to  r\.  If 
AD  and  BC  were  two  equal,  independent  circular  disks,  each 
would  have  true  rolling  motion,  and  BC  would  make  r\  revolu- 
tions, while  AD  would  make  r%.  But  BC  and  AD  are  both 
disks  of  the  same  roller,  R,  and  cannot  rotate  relative  to  each 
other;  hence  they  must  each  make  the  same  number  of  revo- 
lutions, and  points  C  and  D  of  the  disks  would  have  to  have 
the  same  velocity,  which  is  inconsistent  with  the  conditions  of 
motions  of  C  and  D  as  points  of  S.  Hence  a  roller  cannot 
be  correctly  used  for  a  thrust-bearing.  Short  rollers  securely 
held  in  cages  are  used  in  practice,  but  experiments  show  that 
they  are  not  as  efficient  as  properly  designed  forms.* 

Consider  case  C.  Relative  to  T,  the  double  point  D  will 
have  a  linear  velocity  proportional  to  r2  and  C  will  have  a 
linear  velocity  proportional  to  r\.  Consider  AD  and  BC  as 

RC*     Y 

independent  disks  so  proportioned  that  -rjz  =— .     If  D  has  a 


*  See  article  by  T.  Hill  in  American  Machinist,  Jan.  5,  1899.     Also  description 
of  bearing  by  C.  R.  Pratt,  same  periodical,  June  27,  1901. 


ROLLER-  AND  BALL-BEARINGS.  267 

linear  velocity  proportional   to    r2,  then   the    angular   velocity 
of  AD   about    its    axis   OX  will   be    proportional   to 


Similarly,  the  angular  velocity  of    EC  about  axis  OX  will  be 
proportional  to 


since 


Angular  velocity  of  A D     nAD      BC  r^    r\ 
Angular  velocity  of  BC        r\        AD  TI     r2 

iiBC 
BC    rv 


AD    r2' 

Hence  the  disks  AD  and  BC  have  the  same  angular  velocity 
about  the  axis  OX,  and  may  form  parts  of  the  same  body. 
This  will  be  true  of  any  pair  of  disks  of  the  cone  OBC.  Any 
frustum  of  a  cone  whose  apex  lies  anywhere  on  the  axis  OY 
will  therefore  fulfill  the  conditions  for  true  rolling  motion  rela- 
tively to  T  when  actuated  by  S. 

In  each  of  the  foregoing  cases  the  rolling  members  must  be 
held  in  suitable  "  cages,"  or  they  will  yield  to  the  tendency  to 
displace  them. 

In  ball  thrust-bearings  it  is  desirable  to  so  arrange  the  balls 
in  the  cage  that  each  one  will  have  a  separate  path,  as  this 
minimizes  wear. 

For  a  three-point  thrust  ball-bearing  the  form  of  the  races 
to  permit  true  rolling  must  be  as  shown  in  Fig.  151  to  be  in 
accordance  with  the  principles  just  demonstrated.  The  groove- 
angle  should  be  as  flat  as  possible  to  reduce  the  friction  effect 
of  "  spinning." 

About  120°  will  be  found  a  good  practicable  value. 

The  ball  becomes  akin  to  a  cone  as  far  as  its  relations  with 
T  and  S  are  concerned.  In  each  case  the  motion  imparted  to 


268 


MACHINE  DESIGN. 


the  ball  tends  to  rotate  it  about  the  correct  axis  OX  and  the  con- 
ditions for  true  rolling  are  satisfied.  The  sides  of  the  race  are 
tangent  to  the  ball  where  it  is  cut  by  any  line  A-B  which  passes 
through  O.  Boring  at  A  and  B  is  unavoidable. 

A    four-point    ball-bearing   must    be    designed    according   to 
the  principles  indicated  in  Fig.  152  for  true  rolling  motion.     As 


FIG.  152. 


FIG.  153. 


far  as  its  motion  relations  with  T  and  S  are  concerned,  the  ball 
becomes  akin  to  a  cone.*  Boring  occurs  at  all  four  tangent  points. 

Similarly  a  cup  and  cone  three-point  bearing  should  have 
the  form  shown  in  Fig.  153. 

146.  II.  The  form  of  the  constraining  surfaces  must  be  such 
in  ball-bearings  that  the  balls  will  not  have  any  effective  tendency 
to  leave  their  proper  paths.  The  use  of  cages  for  this  purpose 
has  already  been  mentioned. 

*  The  method  of  laying  out  the  groove  in  P'ig.  151  is  as  follows :  —  The  axes  of 
rotation  of  the  balls  cut  the  main  axis  of  the  bearing  at  O.  Draw  the  lower  surface 
of  5  tangent  to  the  ball  at  C  and  parallel  to  the  ball  axis  OX.  Draw  the  line  OB 
cutting  the  ball  at  A  and  B,  and  draw  tangent  surfaces  normal  to  the  radii  of  the 
ball  at  A  and  B.  These  surfaces  form  the  groove  angle  BDA.  If  the  first  trial 
gives  too  sharp  a  groove  angle,  increase  the  angle  XOB  and  repeat  the  construction. 
If  BDA  is  too  flat,  decrease  XOB. 

For  the  four-point  bearing  shown  in  Fig.  152  the  same  method  is  used  for  deter- 
mining the  groove  in  5  as  well  as  T. 


ROLLER-  AND  BALL-BEARINGS. 


269 


If  two-point  bearings  without  cages  are  desired,  the  section 
of  each  race  should  be  the  arc  of  a  circle  whose  radius  is  ^ 
to  J  of  the  diameter  of  the  ball.  -In  two-point  bearings  the 
points  of  pressure  must  always  be  diametrically  opposite  except 
as  noted  for  thrust-bearings. 

In  three-  and  four-point  bearings  where  the  races  are  properly 
formed  for  true  rolling,  as  explained  in  the  preceding  section,  the 
tendency  for  the  balls  to  leave  the  races  is  reduced  to  a  minimum. 

One  point,  however,  needs  further  consideration.  In  cup- 
and  cone-bearings  it  is  impossible  to 
keep  a  tight  adjustment  at  all  times, 
and  the  least  play  will  allow  some  of 
the  balls  on  the  unloaded  side  of  the 
bearing  to  get  out  of  place. 

Fig.    154   shows    such    a     bearing 
loosely  adjusted. 

The  loaded  cup  is  forced  down  so 
that  its  axis  lies  below  the  axis  of 
the  cone.  The  top  ball  is  held  correctly  in  place  for  true 
rolling;  the  lower  ball  is  free  to  roll  to  one  side  as  seen. 
Investigation  has  shown  that  the  angles  a  and  /?  should  each 
be  at  least  as  great  as  25°  in  order  to  return  the  displaced  ball 
easily  to  its  proper  path  by  the  time  it  becomes  subjected  to 
the  load.  If  the  angles  are  too  acute  there  is  a  tendency  for 
the  balls  to  wedge  in  their  incorrect  positions,  causing  rapid 
wear  or  even  crushing.* 

147  III.  Allowable  Loading. — Careful  experiments  show 
that  for  high  efficiency  and  durability  the  loads  on  balls  and 
rollers  should  be  very  much  less  than  they  could  be  with  safety 
as  far  as  their  strength  is  concerned.! 

*  See  article  by  R.  Janney  in  American  Machinist,  Jan.  5,  1899. 
t  See  excellent  article  by  Professor  Stribeck  in  Z.  d.  V.  d.  I.,  Jan.  19  and  26, 
1901. 


FIG.  154. 


270  MACHINE  DESIGN. 

Let  PO  equal  the  load  in  pounds  which  is  allowable  for  a  single 
ball  or  roller.     Then  for  balls 


d=  diameter  of  ball  in  inches. 

K  =  1  500  for  hardened  steel  balls  and  races,  two-point  bearing, 

with  circular-arc  races  having  radii  equal  to  f  d. 
K=    750  for  hardened   steel   balls  and    races,  three-  and  four- 

point  bearing. 

For  two-point  bearing  with  flat  races,  K  =  500. 
For  cast-iron  balls  and  races  use  two-fifths  of  these  values. 
For  bearings  in  which  the  greatest  care  has  been  taken  re- 
garding the  selection  of  the  most  suitable  steel,  its  proper  heat 
treatment,  and  accuracy  of  workmanship,  these  values  may  be 
increased  50  per  cent. 
For  rollers  P0  =  Kdl. 

d=  diameter  of  roller  in  inches  =  mean  diameter  of  cone, 
/  =  length  of  roller  in  inches, 
K  =  400  for  cast  iron, 
K  =  1000  for  hardened  steel. 
In  thrust-bearings,  if  the  total  load  =  P  and  the  number  of 

P 

balls  —  ny  we  have  for  either  balls  or  rollers  P0  =—  . 

In  cylindrical  bearings  the  load  is  always  greatest  at  one 
side  of  the  bearing,  the  balls  or  rollers  on  the  opposite  side 
being  entirely  unloaded.  It  has  been  found  that  the  load  on 

the  heaviest   loaded   ball  or  roller  =  P0  =  —  P,   where  n  is   the 

ft 

number  of  balls.* 


*  Mr.  Henry  Hess,  of  the  Hess-Bright  Mfg.  Co.,  in  a  letter  to  the  authors  says:  — 
"  It  is  a  fact,  that  has  been  determined  by  experience,  that  in  radial  [i.e.,  cylindrical] 
ball  bearings  the  speed  has  very  little  influence  within  very  wide  limits.  In  my 
practice  ...  I  pay  no  attention  to  speed  of  radial  bearings  up  to  3000  rpm  as 


ROLLER-  AND  BALL-BEARINGS. 


27I 


For  radial  bearings,  not  subjected  to  shock,  speed  up  to  3000 
r.p.m.  need  not  affect  the  allowable  load. 
For  thrust  ball-bearings, 


P= 


$ooond2 


2.55 


.m. 


for  best  material  and  workmanship  and  circular  arc  races.     For 
cast  iron  or  soft  steel  divide  by  100. 

148.  Size    of    Bearing. — To  determine  the    size  of  the  ball 


influencing  the  load  so  long  as  such  speed  is  fairly  uniform  and  so  long  as  the  load 
is  fairly  uniform.  When  neither  speed  nor  load  are  uniform  the  percussive  effect 
of  rapid  changes  must  be  taken  in  consideration;  unfortunately,  so  far  at  least,  the 
factors  are  entirely  empirical  and  allowances  are  made  by  a  comparison  with 
analogous  cases  of  previous  practice. 

The  case  is  different  with  thrust  bearings.  In  these,  speed  is  a  very  decided 
factor  in  the  carrying  capacity  even  though  speed  and  load  be  uniform.  Here 
again  no  rational  formula  has  yet  been  developed  to  adequately  represent  the 
different  elements,  but  carrying  capacities  for  different  speeds  of  standard  bearings 
have  been  experimentally  determined,  since  it  was  quite  feasible  to  get  different 
uniform  speeds  and  determine  under  what  load  the  carrying  capacity  was  reached. 
We  found,  for  instance,  that  for  a  thrust  bearing  employing  18  —  \"  balls,  the 
permissible  load  at  10  rpm  was  2400  pounds;  at  300  rpm  —  650  pounds;  at  iooo  — 
450  pounds;  and  at  1500  only  330  pounds.  We  also  found  that,  generally  speaking, 
it  was  inadvisable  to  use  this  type  of  bearing  for  speeds  materially  above  1500  rpm." 

An  analysis  of  certain  standard  thrust  bearings  in  connection  with  the  makers' 
catalog  allowances  for  loads,  gives  at  various  speeds:  — 

LOAD  PER  BALL,  POUNDS. 


R.  P.  M. 

1/4"  Ball. 

5/1  6"  Ball. 

3/8"  Ball. 

1500 

IOOO 

500 
300 
150 

10 

27.8 

33-3 
41.7 
55-6 
61.2 

210 

35.4-44-7 
4I-7-57-I 
52.1-71.4 
66.7-89.2 
83-3-io7 
229-339 

80.7 
91.7 
129 
153 
193 
56o 

C.  R.  Pratt,  Trans.  A.S.M.E.,  Vol.  27  gives  as    limiting  load  per  \"  ball,  100 
pounds  at  700  rpm  with  a  6"  diameter  circle  of  rotation. 


272 


MACHINE  DESIGN. 


circle  (i.e.  the    middle    diameter    of    the     <k  race  ")    given    the 
number  of   balls  n  and  their   diameter  d.     (See  Fig.  155.) 

r=  radius  of  ball.  R  =  radius  of  ball 
circle.  Join  the  centers  of  two  consec- 
utive balls  by  the  chord  AB  =  2r.  From 
the  center  of  the  ball  circle,  O,  draw  two 
radii,  one  to  A  and  the  other  to  the  mid- 
point of  A-B.  Call  the  angle  included 
between  the  radii  a.  Then 


FIG.  155. 


since 


r  =  Rsma,    and, 
180° 


a 


R  = 


180°' 


sm 


This  is  the  radius  of  a  circle  on  which  the  centers  of  the 
balls  will  lie  when  their  surfaces  are  all  in  contact.  It  is  desir- 
able to  allow  some  clearance  between  the  balls.  This  may 
be  as  much  as  0.005  inch  between  each  pair  of  balls  provided 

the  total  allowance  does  not  exceed  — .      When  the  total  clear- 

4 

ance  has  been  decided  upon,  it  may  be  allowed  for  by  making 
the  actual  radius  of  the  ball  circle  larger  than  R  by  an  amount 
one  sixth  of  the  total  clearance  desired. 

The  most  satisfactory  service  seems  to  be  given  by  those 
radial  bearings  which  employ  some  type  of  elastic  ball  sep- 
arators. 

149.  IV.  Lubrication  and  Sealing. — On  account  of  "  spin- 
ning," faulty  adjustment,  and  unavoidable  slippage,  rolling 
bearings  should  be  properly  lubricated.  As  they  are  extremely 
sensitive  to  the  presence  of  dust  and  grit,  care  must  be  exer- 


ROLLER-  AND  BALL-BEARINGS.  273 

cised  that  the  lubricant  be  admitted  without  any  danger  of  the 
entrance  of  these. 

Sealed  oil-holes,  dust-caps,  and  felt  washers  are  commonly 
used  both  to  retain  the  lubricant  for  bath  lubrication  and  for 
keeping  out  all  dirt. 

150.  V.  Prevention  of  Binding. — Many  ball  bearing  instal- 
lations, otherwise  properly  designed,  have  failed  because  they 
neglected  to  take  into  account  the  fact  that  no  material  is  utterly 
rigid    or    workmanship    mathematically    accurate.     For    these 
reasons  a  bearing  employing  a  double  row  of  balls  so  arranged 
as  to  ignore  accommodation  to  natural  elastic  yielding  will  not 
carry  twice  the  load  of  a  similar  bearing  with  one  row  of  balls. 
Binding  will    inevitably  result  where   no  provisions  for  elastic 
yielding  have  been  made.     The  accommodation  to  temperature 
changes  must  also  be  considered  in  some  installations. 

Manufacturers  issue  valuable  data  sheets  upon  these  matters 
and  invite  consultation  as  to  details  of  installation. 

151.  Efficiency  of  Ball-  and  Roller-bearings. — Although  the 
efficiency  of  these  bearings,  as  shown  by  Stribeck,*  Thomas,  f 
and  others,  shows  a  variation  with  changes  of  load,  velocity,  and 
temperature,  these  variations  are  within  a  relatively  small  range 
for  properly  designed  and  installed  bearings.     The   coefficient 

Fr 

of  friction,  referred  to  the  force  at  the  shaft  =  n  =  — ,  where  Pr  =  the 

turning  moment  exerted  on  the  shaft  and  Fr  the  corresponding 
friction  moment,  r  being  the  shaft  radius. 

For  roller  bearings  /*  ranges  in  value  from  0.0035  to  0.02, 
the  higher  value  corresponding  to  great  underloading.  For 
roller  bearings  properly  proportioned  to  their  load,  a  mean  value 
of  n  =  0.005  mav  be  used. 

For  ball  bearings,  properly  proportioned  and  correctly  in- 
stalled, a  mean  value  of  /*  =  0.002  may  be  used. 

*  Z.  d.  V.  d.  I.,  1901  and  1902.  f  Trans.  A.  S.  M.  E.,  1913. 


CHAPTER  XIV. 

COUPLINGS  AND   CLUTCHES. 

152.  Couplings   and  Clutches  Defined. — Couplings  are  those 
machine  parts  which  are  used  to  connect  the  ends  of  two  shafts 
or  spindles  in  such  a  manner  that  rotation  of  the  one  will  pro- 
duce  an  identical  rotation  of  the  other.     They  are  therefore 
in  the  nature  of  fastenings,  and  may  be  classified  as  perma- 
nent or  disengaging.     The  latter  are  frequently  called  clutches. 

153.  Permanent     Couplings. — The    simplest    form    of    per- 
manent  coupling  is   shown  in  Fig.  156,  and   is    known  as  the 
"  sleeve  "  or  "  muff  "  coupling.     Each  shaft  has  a  keyway  cut 


FIG.  156. 

at  the  end.  The  cast-iron  sleeve  of  the  proportions  indicated  is 
bored  an  exact  fit  for  the  shafts  and  has  a  keyway  cut  its  entire 
length.  When  the  sleeve  is  slipped  over  the  ends  of  the  shafts, 
the  key  is  driven  home  and  all  relative  rotation  is  prevented. 
The  key  may  be  proportioned  according  to  the  rules  kid  down 

in  Chapter  IX. 

274 


COUPLINGS   AND   CLUTCHES. 


275 


154.  Flange    couplings    are    frequently  used,  and  Fig.    157 
illustrates   the   type.     Approximate   proportions   are    indicated. 


FIG.  157. 

The  number  of  bolts  n  may  be  from  3  +  0.5^  to  3  +  d.  Their 
diameter  d'  must  be  such  that  their  combined  strength  to  resist 
a  torsional  moment  about  the  axis  of  the  shaft  will  be  equal 
to  the  torsional  strength  of  the  shaft, 


/•  =  allowable    stress    in   outer   fiber   of  shaft,   pounds   per 

square  inch; 

d  =  diameter  of  shaft,  inches; 

./£  =  radius  of  bolt  circle,  inches;  approximately  1.5^; 
n  =  number  of  bolts;  usually  an  even  number,  4,  6,  8,  etc.; 
d'  =  diameter  of  bolts,  inches; 

//  =  allowable  shearing  stress  in  bolts,  pounds  per  square  inch. 
This  equation  will  approximately  give 


155.  Compression    couplings  of  three  forms  are  shown  in 
Figs.  158,  159,  and   160.     The  first  is  similar  to  the  ordinary 


276 


MACHINE  DESIGN. 


flange  coupling  except  that  the  flanges  draw  up  on  a  sleeve  which 
is  split  in  halves  longitudinally  and  is  tapered  toward  each  end 


FIG.  158. 

on  the  outside.     The  two  flanges  have  internal  tapered  surfaces 
to  fit  these. 


Tit  Key 


FIG.  159. 

Instead  of  being  held  by  rings  the  half  sleeves  are  sometimes 
bolted  together  as  shown  in  Fig.  159. 


gection  a-b 


FlG.  160. 


FlG.  i6oA, 


156.  The  "Sellers"  coupling  is  shown  in  Fig.  160.  An 
outer  sleeve  A  is  bored  tapering  from  each  end.  A  split  cone 
bushing  B  is  inserted  at  each  end.  Openings  are  left  for  three 


COUPLINGS  AND  CLUTCHES. 


277 


bolts  by  means  of  which  B  and  B  are  drawn  toward  each  other 
and  thus  closed  down  on  the  shaft  with  great  force.  A  key 
is  also  provided.  The  outer  sleeve  may  be  made  in  two  parts 
with  an  interlocking  joint,  Fig.  i6oA,  so  that  the  coupling  can 
be  used  without  slipping  the  sleeve  on  from  the  end  of  the 
shaft. 

157.  Oldham's    Coupling. — For   all    of   the   foregoing  coup- 
lings the  axes  of  the  two  shafts  must  be  identical.     Where  the 

axes  are  parallel,  Oldham's  coupling 
may  be  used.  It  is  shown  in  Fig.  161, 
and  consists  of  three  parts.  Each  shaft 
end  has  keyed  to  it  a  flange  or  disk 
which  has  a  diametral  groove  cut  across 
its  face.  Between  these  two  disks  is 
a  third  which  has  a  tongue  on  each  face.  The  tongues,  which 
just  fit  the  grooves  freely,  run  diametrically  across  the  disk 
and  are  at  a  right  angle  with  each  other. 

158.  Hooke's    Coiipling. — For  axes  which  intersect,  Hooke's 
coupling  or  "  universal  joint  "  may  be  used.     It  is  shown  in 
outline  in  Fig.  162.     Each  shaft  has  a  stirrup  keyed  to  its  end. 


FIG.  161. 


FIG.  162. 


Each  stirrup  is  connected  by  turning  pairs  to  a  cross-shaped 
intermediate  member,  the  axes  of  whose  turning  pairs  are 
at  a  right  angle.  For  a  mathematical  analysis  of  this  mechanism 


278 


MACHINE  DESIGN. 


the  reader  is  referred  to  Professor  Kennedy's  "  Mechanics  of 
Machinery." 

159.  Hobson's  Coupling.* — A  novel  coupling  for  transmitting 
motion   from   axes   at   90°   at   equal,    uniform   angular   velocity 
(equivalent  to  a  pair  of  miter  gears)  is  known  as  Hobson's  Coup- 
ling.    As  seen  in  Fig.  162^!,  the  coupling  heads  are  merely  cyl- 
inders,  bored  for  and  keyed   to   their  respective  shafts.     Near 

their  circumference  and  evenly 
spaced,  holes  are  drilled  to  receive 
comfortably  a  number  of  rods. 
These  rods  are  bent  to  exactly  90° 
and  each  leg  has  a  length  of  a  coup- 
ling head  plus  the  shortest  expos^i 
length  as  shown  at  B.  They  are 
free  to  turn  in  their  sockets  and  to 
slide  lengthwise  as  the  relative 
movement  of  the  heads  demands. 
When  in  the  extreme  position  A, 

the  ends  of  a  rod  are  midway  in  the  heads,  and  in  position  B 
are  flush  with  the  outer  faces.  The  device  should  be  enclosed 
in  a  case  and  well  lubricated.  The  area  of  the  rods  should  be 
made  sufficient  to  equalize  their  carrying  power  to  the  torsional 
strength  of  the  shaft. 

160.  Flexible  Couplings. — Where  shafts  which  are  or  which 
may  become   slightly   out   of   alignment   are   to   be   connected, 
some   form   of  flexible   coupling  is  advisable.     Their  principle 
is  illustrated  in  Fig.   163.     Each  shaft  has  keyed  to  its  end  a 
disk  which  has  set  in  its  face  a  number  of  pins.     The  pins  are 
so  placed  that  those  in  the  one  circle  will  not   strike  those  in 
the  other  if  either  shaft  is  rotated  while  the  other  remains  at 
rest.     When  one   shaft   is   to  drive   the   other,   short   belts   are 


FIG.  162,4. 


*  Hiscox's  Mechanical  Movements.     See  also  Power,  Jan.  26,  1915. 


COUPLINGS  4ND   CLUTCHES. 


279 


placed  on  the  pins  as  shown  in  B,  Fig.  163.  The  same  general 
idea  is  used  in  a  coupling  employing  a  single  continuous 
belt. 

Another  device  for  the  same  purpose  is  one  which  employs 
a  flexible  disk,  shown  in  Fig. 


FIG.  163. 


FIG.  1634. 


161.  Disengaging  couplings  are  of  two  general  classes: 
positive  drive  and  friction  drive.  Positive-drive  couplings  are 
commonly  called  toothed  or  claw  couplings.  They  consist 
of  two  members  having  projections  on  their  faces,  as  shown 
in  Fig.  164,  which  interlock  when  in  action.  A  is  keyed  rigidly 


FIG.  164. 

to  its  shaft.     B  can  slide  along  its  shaft  guided  by  the  feather,  F, 
but  cannot  rotate  except  with  the  shaft.      When  B  is  moved 


280  MACHINE  DESIGN. 

in  the  direction  of  the  arrow,  its  teeth  engage  with  those  of  A, 
and  the  two  shafts  must  have  the  same  motion  of  rotation. 
D  shows  the  end  of  the  lever  which  moves  B.  The  split  ring 
is  bolted  around  B  in  the  groove  E,  which  it  fits  freely.  Some- 
times short  blocks  which  fit  E  are  used  in  place  of  the  entire 
ring.  C  shows  an  end  view  of  the  half  clutch. 

Various  forms  of  teeth  may  be  used.*  If  n  equals  the  number 
of  teeth  and  R  equals  the  mean  radius  of  the  clutch  tooth  the 
following  equations  may  be  written  : 


and  }J~-=RnAf9', 


In   the  first   equation,   fs~7~)   the   torsional  strength  of  the 

shaft,  is  equated  to  the  crushing  resistance  of  all  the  teeth 
opposed  to  the  torsional  stress.  A'  is  the  area  of  the  en- 
gaging face  of  one  tooth  and  }c  the  allowable  unit  crushing 
stress. 

In  the  second  equation  A  is  the  area  of  the  root  of  the  tooth 
subjected  to  shear  and//  is  the  allowable  unit  shearing  stress. 

162.  Friction  couplings  f  generally  consist  of  two  parts,  a 
hollow  cone,  A,  keyed  rigidly  to  one  shaft,  and  a  sliding  cone, 
B}  held  by  a  feather  on  the  second  shaft  as  seen  in  Fig.  165. 
By  means  of  a  lever,  B  can  be  forced  against  A  with  a  con- 
siderable axial  pressure.  This  induces  friction  between  the 
conical  surfaces,  which  friction  resists  relative  rotation.  To 
analyze  the  forces,  consider  Fig.  166  which  shows  two  conical 

*  See  article  by  G.  H.  Marx,  in  American  Machinist,  July  9,  1903. 
f  For  valuable  material  see  paper  by  Henry  Souther,  Trans.  A.  S.  M.  E., 
Vol.  30,  from  which  some  of  the  following  illustrations  are  copied. 


COUPLINGS  AND   CLUTCHES. 


281 


surfaces  pressed  together  by  an  axial  force  P.    The  angle  at  the 
vertex  of  the  cone  is  2a. 


FIG.  165.  FIG.  166. 

Let  the  coefficient  of  friction  =  /*  =  tan  <£. 
The  mean  cone  radius  =r. 

Total   pressure    between    the    two    surfaces   for   impending 
slippage  =2R. 

Total  normal  pressure  between  two   surfaces  at  rest=2AT. 

Total  friction  =  F  =  ^N. 

It  is  clear  from  the  figure  that  P=2Rsm  (« 

Also,  N  =R  cos  (j>. 

2N  sin  (a  +  0 ) 
cos  <> 


.'.  p 


But  2N  = — ,  where  F=  friction  force  between  the  surfaces. 

F 

:.  P= — (sin  a  +  ficos  a). 

From  this  equation  and  the  fact  that  the  turning  moment, 
M  =Fr,  the  clutch  can  be  designed  to  transmit  the  desired  power. 

The  angle  a  should  lie  between  7^°  and  12^°  in  order  to  avoid 
"  sticking"  on  the  one  hand  and  too  sudden  seizure  on  the  other. 

The  normal  pressure  per  sq.  in.  of  contact  area  may  be  taken 
as  30-40  Ibs.  for  leather  or  cork,  and  35-4^  Ibs.  for  maple  wood. 


282 


MACHINE  DESIGN. 


TABLE  XVIII. 

ji=o.io  to  0.15  for  cast  iron  on  cast  iron 
=  0.15  to  0.20  for  cast  iron  on  paper 
=  0.20  to  0.30  for  cast  iron  on  leather 
=  0.20  to  0.40  for  cast  iron  on  wood 
=  0.33  to  0.37  for  cast  iron  on  cork. 

The  range  in  values  of  /*  is  due,  not  only  to  surface  differences, 
or  the  presence  of  various  amounts  of  lubricating  matter,  but 
also  to  variety  in  the  rate  of  slippage.  With  metal  on  metal  fj. 
decreases,  from  its  value  as  the  coefficient  of  friction  of  rest, 
with  increase  of  velocity  of  slip.  But  the  reverse  is  true  with 
leather. 

It  is  obvious,  in  the  elementary  cone  clutch  of  Fig.  165,  that 
the  end  thrust  P  produces  an  undesirable  and  excessive  thrust- 


FIG. 


FIG.  1 665. 


bearing  friction  on  each  shaft.  For  this  reason  the  self-sustaining 
principle  is  used.  The  load  may  be  applied  by  a  spring  as 
in  Fig.  i66^4,  or  by  an  adjustable  self-locking  thrust-block  device, 
as  in  Fig.  i66B,  but  in  either  case  there  is  no  external  end-thrust 
when  the  clutch  is  driving. 


COUPLINGS  AND  CLUTCHES. 


283 


163.  Weston  Friction  Coupling. — For  heavy  duty  the  prin- 
ciple of  the  Weston  friction  coupling,  as  shown  in  Fig.  167,  may 
be  used.  The  sleeve  A  carries  two  feathers  on  which  a  number 


Section  EJJ 


FIG.  167. 


of  iron  rings  C  can  slide  but  not  rotate.  Similarly  the  hollow 
sleeve  B  is  provided  with  feathers  which  prevent  the  rotation 
of  the  wooden  rings  D,  while  not  interfering  with  their  sliding. 


FIG.  1674, 

Let  there  be  n  iron  rings.  Then,  when  B  is  pressed  toward 
Ay  there  will  be  friction  induced  on  2n  +  i  surfaces.  If  P  is 
the  axial  pressure  and  //  the  coefficient  of  friction,  the  total 
friction  F  =  nP (2n  + 1) ,  and  if  r=the  mean  radius  of  the  rings, 
the  moment  which  can  be  transmitted  =  M  =  Fr. 

This  clutch  belongs  to  the  class  known  as  disk  clutches.     They 


284 


MACHINE  DESIGN. 


exist  in  great  variety,  with  single  or  multiple  disks,  flat  or  V 
shaped. 

164.  Band  and  Other  Friction  Clutches. — Band  clutches 
use  a  flexible  member  and  operate  somewhat  on  the  principle 
of  friction  brakes.  (See  Chap.  XV.)  They  are  made  of  both 
expanding  and  contracting  band  types,  shown  in  Figs.  167^.  and 
167$  respectively. 

The  same  general  idea  is  used  in  a  great  variety  of  friction 
clutches  employing  radially  adjustable  blocks  which  close  upon 


FIG.  1675. 

a  solid  band  or  ring.  Ordinarily  the  blocks  are  pressed  against 
the  ring  by  the  action  of  right-  and  left-hand  screws  or  toggle 
devices.  Illustrations  are  plentiful  in  trade  catalogues. 

.  A  combination  friction-  and  claw-clutch  is  shown  in  Fig. 
1 68.  To  start  the  driven  shaft,  B  is  forced  to  the  right,  thus 
bringing  the  friction  cones  into  action.  When  the  driven  shaft 
has  attained  its  proper  speed,  B  is  suddenly  shifted  to  the  left, 
thus  causing  the  claw-clutch  to  engage,  which  gives  the  ad- 
vantage of  positive  driving. 

Professor  Bach  has  shown  that  the  energy  lost  in  friction  in 
making  a  "  running  start  "  with  a  friction  clutch  just  equals 


COUPLINGS  AND  CLUTCHES. 


'85 


the  kinetic  energy  of  the  shaft  and  attached  parts  at  the  speed 
to  which  they  have  been  brought.  This  shows  clearly  the 
enormous  wear  and  tear  to  which  friction  clutches  are  ordinarily 
subjected,  even  when  designed  to  run  practically  without  slip  in 
the  course  of  operation,  and  explains  their  rapid  deterioration. 


FIG.  168. 

For  power  house  purposes  very  satisfactory  magnetic  clutches 
have  been  devised.  The  most  serious  objection  raised  against 
them  has  been  due  to  their  suddenness  of  seizure  unless  carefully 
controlled. 

Pneumatic  clutches  of  the  disk  type  have  been  successfully 
employed. 

Probably  the  most  promising  field  of  clutch  development 
lies  in  the  direction  of  hydraulic  clutches,*  as  these  offer  a  means 
of  obtaining  various  speed  ratios,  coupled  with  the  advantages 
of  very  gradual  seizure  and  almost  entire  freedom  from  wear. 

*  Engineering  News,  Aug.  10,  1911,  p.  164. 


CHAPTER   XV. 
BELTS,  ROPES,  BRAKES,  AND   CHAINS. 

165.  Transmission  of  Motion  by  Belts.— In  Fig.  169,  let  A 
and  B  be  two  cylindrical  surfaces,  free  to  rotate  about  their  axes; 
let  CD  be  their  common  tangent,  and  let  it  represent  an  inex- 
tensible  connection  between  the  two  cylinders.  Since  it  is 
inextensible,  the  points  D  and  C,  and  hence  the  surfaces  of  the 


cylinders,  must  have  the  same  linear  velocity  when  A  is  rotated 
as  indicated  by  the  arrow.  Two  points  having  the  same  linear 
velocity,  and  different  radii,  have  angular  velocities  which  are 
inversely  proportional  to  their  radii.  Hence,  since  the  surfaces 
of  the  cylinders  have  the  same  linear  velocity,  their  angular 
velocities  are  inversely  proportional  to  their  radii.  This  is  true 
of  all  cylinders  connected  by  inextensible  connectors.  Suppose 
the  cylinders  to  become  pulleys,  and  the  tangent  line  to  become 
a  belt.  Let  C'D'  be  drawn;  this  becomes  a  part  of  the  belt 
together  with  the  portions  DED'  and  CFC',  making  it  endless, 
and  rotation  may  be  continuous.  The  belt  will  remain  always 
tangent  to  the  pulleys,  and  will  transmit  such  rotation  that  the 

286 


BELTS,  ROPES,  BRAKES,  AND  CHAINS.  287 

angular  velocity  ratio  will  constantly  be  the  inverse  ratio  of  the 
radii  of  the  pulleys. 

The  case  considered  corresponds  to  a  crossed  belt,  but  the 
same  reasoning  applies  to  an  open  belt.    (See  Fig.  170.)    A  and 


B  are  two  pulleys,  and  CDD'C'C  is  an  open  belt.  Since  the 
points  C  and  D  are  connected  by  a  belt  that  is  practically  inex- 
tensible,  the  linear  velocity  of  C  and  D  is  the  same;  therefore 
the  angular  velocities  of  the  pulleys  are  to  each  other  inversely 
as  their  radii.  If  the  pulleys  in  either  case  were  pitch  cylinders 
of  gears  the  condition  of  velocity  would  be  the  same.  In  the 
first  case,  however,  the  direction  of  motion  is  reversed,  while 
in  the  second  case  it  is  not.  Hence  the  first  corresponds  to 
gears  meshing  directly  with  each  other,  while  the  second  corre- 
sponds to  the  case  of  gears  connected  by  an  idler,  or  to  the  case 
of  an  annular  gear  and  pinion.  While  in  many  places  positive 
driving- gears  are  indispensable,  it  is  frequently  the  case  that  the 
relative  position  of  the  axes  to  be  connected  is  such  as  would 
demand  gears  of  inconvenient  or  impossible  proportions,  and 
belts  are  used  with  the  sacrifice  of  positive  driving. 

Of  course  it  is  necessary  that  a  belt  should  have  some  thick- 
ness; and,  since  the  center  of  pull  is  the  center  of  the  belt,  it 
is  necessary  to  add  to  the  radius  of  the  pulley  half  the  thickness 
of  the  belt.  The  motion  communicated  by  means  of  belting, 
however,  does  not  need  to  be  absolutely  correct,  and  therefore 
in  practice  it  is  usually  customary  to  neglect  the  thickness  of  the 


288  MACHINE  DESIGN. 

belt.     The  proportioning  of  pulleys  for  the  transmission  of  any 
required  velocity  ratio  is  now  a  very  simple  matter. 

166.  Illustration. — A  line-shaft  runs  150  revolutions  per  min- 
ute, and  is  supported  by  hangers  with  16  inches  "drop."  It  is 
required  to  transmit  motion  from  this  shaft  to  a  dynamo  to  run 
1800  revolutions  per  minute.  A  30-inch  pulley  is  the  largest 
that  can  be  conveniently  used  with  1 6-inch  hangers.  Let 
#  =  the  diameter  of  required  pulley  for  the  dynamo;  then  from 
what  has  preceded  x  +  30=  150-^- 1800,  and  therefore  #  =  2.5 
inches.  But  a  pulley  less  than  4  inches  diameter  should  not  be 
used  on  a  dynamo.*  Suppose  in  this  case  that  it  is  6  inches. 
It  is  then  impossible  to  obtain  the  required  velocity  ratio  with 
one  change  of  speed,  i.e.,  with  one  belt.  Two  changes  of  speed 


FIG.  171. 

may  be  obtained  by  the  introduction  of  a  counter-shaft.  By 
this  means  the  velocity  ratio  is  divided  into  two  factors.  If 
it  is  wished  to  have  the  same  change  of  speed  from  the  line  shaft 
to  the  counter  as  from  the  counter  to  the  dynamo,  then  each 
velocity  ratio  would  be  V(i8oo-^  150)  =  ^12  =  3.46.  But  this 
gives  an  inconvenient  fraction,  and  the  factors  do  not  need  to  be 

*  This  limiting  size    is   determined  mainly  by  considerations  of  thickness  of 
belt  reauired  to  transmit  the  energy,  its  durability,  and  its  efficiency.     (See  §  178.) 


BELTS,  ROPES,  BRAKES,  4ND   CHAINS.  289 

equal.  Let  the  factors  be  3  and  4.  (See  Fig.  171.)  A  repre- 
sents the  line-shaft,  B  the  counter,  and  C  the  dynamo-shaft. 
The  pulley  on  the  line-shaft  is  30  inches,  and  the  speed  is  to  be 
three  times  as  great  at  the  counter,  therefore  the  pulley  on  the 
counter  connected  with  the  line-shaft  pulley  must  have  a  diam- 
eter one  third  as  great  as  that  on  the  line-shaft  =  10  inches. 
The  pulley  on  the  dynamo  is  6  inches  in  diameter  and  the  counter- 
shaft is  to  run  one  fourth  as  fast  as  the  dynamo,  and  therefore 
the  pulley  on  the  counter  opposite  the  dynamo-pulley  must  be 
four  times  as  large  as  the  dynamo-pulley  =  24  inches. 

167.  A  belt  may  be  shifted  from  one  part  of  a  pulley  to 
another  by  means  of  pressure  against  the  side  which  advances 
towards  the  pulley.  Thus  if,  in  Fig.  172, 
the  rotation  be  as  indicated  by  the  arrow, 
and  side  pressure  be  applied  at  A,  the  belt 
will  be  pushed  to  one  side,  as  is  shown,  and 
will  consequently  be  carried  into  some  new 
position  on  the  pulley  further  to  the  left  as  it 
advances.  Hence,  in  order  that  a  belt  may 
maintain  its  position  on  a  pulley,  THE  CEN- 

FlG.    172. 

TER    LINE     OF    THE    ADVANCING    SIDE    OF  THE 


BELT     MUST     BE     PERPENDICULAR     TO     THE     AXIS    OF    ROTATION. 

When  this  condition  is  fulfilled  the  belt  will  run  and  trans- 
mit the  required  motion,  regardless  of  the  relative  position  of 
the  shafts. 

168.  Twist  Belts.— In  Fig.  173  the  axes  AB  and  CD  are 
parallel  to  each  other,  the  above  stated  condition  is  fulfilled, 
and  the  belt  will  run  correctly;  but  if  the  axis  CD  were  turned 
into  some  new  position,  as  C'D',  the  side  of  the  belt  that  advances 
toward  the  pulley  E  from  F  cannot  have  its  center  line  in  a 
plane  perpendicular  to  the  axis,  AB,  and  therefore  it  will  run  off. 
But  if  a  plane  be  passed  through  the  line  CD,  perpendicular  to 
the  plane  of  the  paper,  then  the  axis  may  be  swung  in  this  plane 


290 


MACHINE  DESIGN. 


in  such  a  way  that  the  necessary  condition  shall  be  fulfilled,  and 
the  belt  will  run  properly.  This  gives  what  is  known  as  a 
"twist"  belt,  and  when  the  angle  between  the  shaft  becomes 
90°,  it  is  a  "quarter-twist"  belt.  To  make  this  clearer,  see 
Fig.  174.  Rotation  is  transmitted  from  A  to  B  by  an  open  belt, 
and  it  is  required  to  turn  the  axis  of  B  out  of  parallelism  with 


E 

A 

B 

/ 

y 

/ 

1 

1 

1 

1 

1     1 

1     l 

1     I 

c'  /    ' 

—  . 

*XN  '    // 

c  JbL 

/ 

D 

$ 

^ 

^D' 

F         fc  — 

FIG.  173. 

FIG.  174. 


that  of  A.  The  direction  of  rotation  is  as  indicated  by  the 
arrows.  Draw  the  line  CD.  If  now  the  line  CD  is  supposed 
to  pass  through  the  center  of  the  belt  at  C  and  D,  it  may 
become  an  axis,  and  the  pulley  B  and  the  part  of  the  belt  FC 
may  be  turned  about  it,  while  the  pulley  A  and  the  part  of  the 
belt  ED  remain  stationary.  During  this  motion  the  center  line 
of  the  part  of  the  belt  CF,  which  is  the  part  that  advances  toward 
the  pulley  B  when  rotation  occurs,  is  always  in  a  plane  perpen- 
dicular to  the  axis  of  the  pulley  B.  The  part  ED,  since  it  has 
not  been  moved,  has  also  its  center  line  in  a  plane  perpendicular 
to  the  axis  of  A.  Therefore  the  pulley  B  may  be  swung  into 
any  angular  position  about  CD  as  an  axis,  and  the  condition 
of  proper  belt  transmission  will  not  be  interfered  with. 

169.  If  the   axes  intersect  the  motion  can  be  transmitted 


BELTS,  ROPES,  BRAKES,   AND   CHAINS. 


291 


between  them  by  belting  only  by  the  use  of  "guide"  or  "idler" 
pulleys.  Let  AB  and  CD,  Fig.  175,  be  intersecting  axes,  and 
let  it  be  required  to  transmit  motion  from  one  to  the  other  by 
means  of  a  belt  running  on  the  pulleys  E  and  F.  Draw  center 
lines  EK  and  FH  through  the  pulleys.  Draw  the  circle,  G, 
of  any  convenient  size,  tangent  to  the  lines  EK  and  FH.  In 
the  axis  of  the  circle,  G,  let  a  shaft  be  placed  on  which  are  two 
pulleys,  their  diameters  being  equal  to  that  of  the  circle,  G. 


FIG.  175. 

These  will  serve  as  guide-pulleys  for  the  upper  and  lower  sides 
of  the  belt,  and  by  means  of  them  the  center  lines  of  the  advanc- 
ing parts  of  both  sides  of  the  belt  will  be  kept  in  planes  perpen- 
dicular to  the  axis  of  the  pulley  toward  which  they  are  advancing, 
the  belts  will  run  properly,  and  the  motion  will  be  transmitted, 
as  required. 

The  analogy  between  gearing  and  belting  for  the  trans- 
mission of  rotary  motion  has  been  mentioned  in  an  earlier 
paragraph.  Spur-gearing  corresponds  to  an  open  or  crossed 
belt  transmitting  motion  between  parallel  shafts.  Bevel-gears 
correspond  to  a  belt  running  on  guide-pulleys  transmitting 
motion  between  intersecting  shafts.  Skew  bevel  and  spiral  gears 
correspond  to  a  "twist"  belt  transmitting  motion  between  shafts 
that  are  neither  parallel  nor  intersecting. 


2Q  2  MACHINE  DESIGN. 

170.  Crowning  Pulleys.  —  If  a  flat  belt  is  put  on  a  "crowning  " 

pulley,  as  in  Fig.  176,  the  tension  on  AB  will  be 
greater  than  on  CD.  The  belt  lying  flat  at  AC  will 
have  its  approaching  portion  bent  out  as  indicated  by 
AE  and  CF,  and  as  rotation  goes  on  the  belt  will 
be  carried  toward  the  high  part  of  the  pulley,  i.e., 
it  will  tend  to  run  in  the  middle  of  the  pulley. 
FIG  176  This  is  the  reason  why  nearly  all  belt  pulleys,  except 
those  on  which  the  belt  has  to  be  shifted  into  different 

positions,  are  turned  "  crowning."     For  pulley  proportions,  see 

Chap.  XVI. 

171.  Cone  Pulleys.  —  In  performing  different  operations  on  a 
machine  or  the  same  operations  on  materials  of  different  degrees 
of  hardness,  different  speeds  are  required.     The  simplest  way 
of  obtaining  them  is  by  use  of  cone  pulleys.     One  pulley  has  a 
series  of  steps,  and  the  opposing  pulley  has  a  corresponding 
series  of  steps.     By  shifting  the  belt  from  one  pair  to  another 
the  velocity  ratio  is  changed.     Since  the  same  belt  is  used  on 
all  the  pairs  of  steps,  these  must  be  so  proportioned  that  the 
belt  length  for  all  the  pairs  shall  be  the  same  ;  otherwise  the  belt 
would  be  too  tight  on  some  of  the  steps  and  too  loose  on  others. 
Let  the  case  of  a  crossed  belt  be  first  considered.     The  length 
of  a  crossed  belt  may  be  expressed  by  the  following  formula: 
Let  L—  length  of  the  belt;   d  =  distance  between  centers  of  rota- 
tion;   R=  radius  of  the  larger  pulley;    r=  radius  of  the  smaller 
pulley.     (See  Fig.  177.)     Then 

L  =  2\/d2-(R  +  r)2  +  (R  +  r)(x  +  2  arc  whose  sine  is  (R  +  r)+d). 


In  the  case  of  a  crossed  belt,  if  the  size  of  steps  is  changed  so 
that  the  sum  of  their  radii  remains  constant,  the  belt  length 
will  be  constant.  For  in  the  formula  the  only  variables  are  R 
and  r,  and  these  terms  only  appear  in  the  formula  as  R  +  r;  but 
R  +  r  is  by  hypothesis  constant.  Therefore  any  change  that  is 


BELTS,  ROPES,  BRAKES,  AND  CHAINS. 


293 


made  in  the  variables  R  and  r,  so  long  as  their  sum  is  constant, 
will  not  affect  the  value  of  the  equation,  and  hence  the  belt 
length  will  be  constant.  It  will  now  be  easy  to  design  cone 
pulleys  for  a  crossed  belt.  Suppose  a  pair  of  steps  given  to 
transmit  a  certain  velocity  ratio.  It  is  required  to  find  a  pair 
of  steps  that  will  transmit  some  other  velocity  ratio,  the  length 
of  belt  being  the  same  in  both  cases.  Let  R  and  r  =  radii  of  the 
given  steps;  Rf  and  /=  radii  of  the  required  steps;  R  +  r  = 
R'Jrrf=a\  the  velocity  ratio  of  Rf  to  r' =b.  There  are  two 


FIG.  177. 

equations  between  R'  and  rf,  Rf  +r'  =b,  and  R'  +  r*  —a.     Com. 
bining  and  solving,  it  is  found  that  r'  =a  +  (i+b),  and  R'  =  a-r\ 
For  an  open  belt  the  formula  for  length,  using  same  symbols 
as  for  crossed  belt,  is 


L  =  2  Vd2  -  (R  -  r)2  +  n(R  +  r) 

—  r]  (arc  whose  sine  is  (R—r)-±-  d). 


If  R  and  r  are  changed  as  before  (i.e., 
constant),  the  term  R  —  r  would  of  course  not  be  constant,  and 
two  of  the  terms  of  the  equation  would  vary  in  value;  therefore 
the  length  of  the  belt  would  vary.  The  determination  of  cone 
steps  for  open  belts  therefore  becomes  a  more  difficult  matter, 
and  approximate  methods  are  almost  invariably  used. 

172.  Graphical  Method  for  Cone-pulley  Design.—  The  fol- 
lowing graphical  approximate  method  is  due  to  Mr.  C.  A.  Smith, 
and  is  given,  with  full  discussion  of  the  subject,  in  "Transactions 


294 


MACHINE  DESIGN. 


of  the  American  Society  of  Mechanical  Engineers,"  Vol.  X, 
p.  269.  Suppose  first  that  the  diameters  of  a  pair  of  cone  steps 
that  transmit  a  certain  velocity  ratio  are  given,  and  that  the 
diameters  of  another  pair  that  shall  serve  to  transmit  some 
other  velocity  ratio  are  required.  The  distance  between  centers 
of  axes  is  given.  (See  Fig.  178.)  Locate  the  pulley  centers  O 


FIG.  178. 

and  Of  at  the  given  distance  apart;  about  these  centers  draw 
circles  whose  diameters  equal  the  diameters  of  the  given  pair  of 
steps;  draw  a  straight  line  GH  tangent  to  these  circles;  at  /, 
the  middle  point  of  the  line  of  centers,  erect  a  perpendicular, 
and  lay  off  a  distance  JK  equal  to  the  distance  between  centers, 
C,  multiplied  by  the  experimentally  determined  constant  0.314; 
about  the  point  K  so  determined,  draw  a  circular  arc  AB  tan- 
gent to  the  line  GH.  Any  line  drawn  tangent  to  this  arc  will  be 
the  common  tangent  to  a  pair  of  cone  steps  giving  the  same 
belt  length  as  that  of  the  given  pair.  For  example,  suppose 
that  OD  is  the  radius  of  one  step  of  the  required  pair;  about  O, 
with  a  radius  equal  to  OD,  draw  a  circle;  tangent  to  this  circle 
and  the  arc  AB  draw  a  straight  line  DE;  about  O'  and  tangent 
to  DE  draw  a  circle j  its  diameter  will  equal  that  of  the  required 
step. 

But  suppose  that,  instead  of  having  one  step  of  the  required 
pair  given,  to  find  the  other  corresponding  as  above,  a  pair  of 


BELTS,  ROPES,  BRAKES,  AND   CHAINS.  295 

steps  are  required  that  shall  transmit  a  certain  velocity  ratio, 
=  r,  with  the  same  length  of  belt  as  the  given  pair.  Suppose 
OD  and  O'E  to  represent  the  unknown  steps.  The  given  velocity 

ratio    equals    r.      Also,  r==^7g-      But  from   similar    triangles 

FO 

OD+O'E=FO^FO'.     Therefore     r=~=-^;    but    FO=C  +  x, 

r(J 

C  +  x  C 

and  FOr  =x.     Therefore  r=~   —,  and  x  = .      Hence  with 

x  r—i 

r  and  C  given,  the  distance  x  may  be  found,  and  a  point  F  located, 
such  that  if  from  F  a  line  be  drawn  tangent  to  AB,  the  cone 
steps  drawn  tangent  to  it  will  give  the  velocity  ratio,  r,  and  a 
belt  length  equal  to  that  of  any  pair  of  cones  determined  by  a 
tangent  to  AB.  The  point  F  often  falls  at  an  inconvenient 
distance.  The  radii  of  the  required  steps  may  then  be  found 
as  follows:  Place  a.  straight-edge  tangent  to  the  arc  AB  and 
measure  the  perpendicular  distances  from  it  to  O  and  O'. 
The  straight-edge  may  be  shifted  until  these  distances  bear 
the  required  relation  to  each  other.  In  this  case  it  is  well  to  check 
the  accuracy  of  the  construction  by  computing  the  resultant 
length  of  belt  with  each  pair  of  steps. 

173.  Design  of  Belts. — Fig.  179  represents  two  pulley? 
connected  by  a  belt.  When  no  moment  is  applied  tending  to 
produce  rotation  this  tension  in  the  two  sides  of  the  belt  is 
practically  equal.  Let  T0  represent  this  tension.  If  now 
an  increasing  moment,  represented  by  Rl,  be  applied  to  the 
driver,  its  effect  is  to  increase  the  tension  in  the  lower  side  of 
the  belt  and  to  decrease  the  tension  in  the  upper  side.  With 
the  increase  of  Rl  this  difference  of  tension  increases  till  it  is 
equal  to  P,  the  force  with  which  rotation  is  resisted  at  the  surface 
of  the  pulley.  Then  rotation  begins  *  and  continues  as  long  as 

*  While  the  moving  parts  are  being  brought  up  to  speed  the  difference  of  ten- 
sion must  equal  P  plus  force  necessary  to  produce  the  acceleration. 


296 


MACHINE  DESIGN. 


this  equality  continues;  i.e.,  as  long  as  TI  -T2  =  P,  in  which 
TI  =  tension  in  the  driving  side  and  T2  =  tension  in  the  slack 
side.  (See  Fig.  180.)  The  tension  in  the  driving  side  is  increased 
at  the  expense  of  that  in  the  slack  side,  but  the  sum  does  not 
remain  a  constant.  Analysis  of  experimental  data  *  shows  that 
a  close  approximation  is  given  by  the  simple  equation 


To  find  the  value  of  ^r.     The  increase  in  tension  from  the 

1  2 

slack  side  to  the  driving  side  is  possible  because  of  the  frictional 
resistance  between  the  belt  and  pulley  surface.     Consider  any 

FIG.  179. 


FIG.  180. 

element  of  the  belt,  ds,  Fig.  181.     It  is  in  equilibrium  under 
the  action  of  the  following  forces: 

T,  the  value  of  the  varying  tension  at  one  end  of  ds; 

T  +  dT,  the  value  of  the  varying  tension  at  the  other  end  of  ds : 

*  Wilfred  Lewis  in  Trans.  A.  S.  M,  E,,  Vol.  VII. 


BELTS,  ROPES,  BRAKES,  AND   CHAINS. 


297 


cds,  the  centrifugal  force; 

pds,  the  pressure  between  the  face  of  the  pulley  and  ds\ 

dF  =  fjipds,  the  friction  between  the  element  of  belt  and  pulley 
face. 

These  correspond  to  any  cross-sectional  area,  A,  square 
inches. 


It  is  more  convenient  to  develop  the  equations  if  a  cross- 
sectional  area  of  one  square  inch  is  considered.     For  such  a 

(T\  /  T*  -L  /7  T\ 

—  },  t  +  dt=  ( — - — j,  cds, 
A/  \     A     / 

pds,  and  fipds. 


295  MACHINE  DESIGN. 


Let  /i  =  —  =  allowable  tension  in  tight  side  of  belt,  pounds 

A 

per  square  inch.     240  Ibs.  is  recommended  as  most 
economical  value. 

T2 

/2  =  —  r  =  tension  in  slack  side,  pounds  per  square  inch; 
j\. 

p 

t\  -/2  =  -j  =  effective  pull,  pounds  per  square  inch; 

A. 

v  =  velocity  of  belt  in  feet  per  second; 
V  =  velocity  of  belt  in  feet  per  minute  ; 
w  =  weight  of  belt,  pounds  per  cubic  inch; 

c  =  centrifugal  force  per  cubic  inch  of  belt  at  velocity 

(I2WV2\ 
-I; 
gr     ' 

r  =  radius  of  pulley  in  inches; 
R  =  radius  of  pulley  in  feet  =  —  ; 

a  =  arc  of  contact  in  degrees; 

0=arc  of  contact  in  radians  =0.0  17  50:; 

T"1 

/0=—  -?  =  initial  tension,  both  sides,  pounds  per  square  inch; 
A 

T 

/c=—  ?  =  centrifugal    tension   per   square    inch   of   cross- 
A 

/I2WV2\ 

section=(-      -I; 
\    g    / 

P  =  pressure  per  linear  inch  between  pulley  and  belt; 
//  =  coefficient  of  friction. 

Summing  the  vertical  components: 

dd  d6 

pds  +  cds  =  t  sin  —  h  (/  +  dt)  sin  —  . 

dd  dO 

dO  is  so  small  that  sin  —  may  be  considered  equal  to  —  . 

2  2 


BELTS,  ROPES,  BRAKES,  AND   CHAINS.  299 

Also  dt  and  dd  being  very  small  compared  to  the  other  quan- 
tities, any  terms  containing  their  product  may  be  dropped. 
Therefore 


but 

I2WV2 


and 


:.   pds=(t~tc)dd. 
Summing  the  moments  about  O  : 


dt  =  [Lpds 

=  p(t-te) 

rti  dt      re 

T—  «/ij   M, 

Jt2  t-tc      Jo 


-tc 

common  log  -1_S  =  .4343^. 

t2  -tc 

The  following  equations  are  now  established 


(i) 


log   —    =  -4343^  ......     (3) 


MACHINE  DESIGN. 


For  speeds  below  1800  ft.  per  minute  equation  (3)  may  be 
written  as 

log  —  =  .4343/*#         (3') 

To  apply  these  equations  it  is  simplest  to  decide  upon  a  max- 
imum value  of  /i.  This  varies  with  the  quality  of  the  belt,  the 
nature  of  the  splice,  etc.,  and  may  be  taken  as  high  as  300  Ibs. 
per  square  inch;  but  when  the  economic  life  of  the  belt  is  con- 
sidered, 200  to  240  Ibs.  per  square  inch  is  better.  For  a  new 
belt  take  240  Ibs. 

'  It  is  evident  that  the  foregoing  equations  may  also  be  written 
in  the  following  forms : 

rT1          D  /       \ 

I  — JL  2==-ij •  .        (Id) 


and  for  low  velocities 


T 
log  ~r=- 

1  2 


The  weight  of  ordinary  oak-tanned  leather  belting  per  cubic 
inch  may  be  taken,  7^  =  0.035. 


TABLE  XIX. 


Values  of  tc  = 


I2WV* 
t 


V 

1800 

2400 

3000 

3600 

4200 

4800 

5400 

6000 

6600 

7200 

7800 

V 

30 

40 

50 

60 

70 

80 

90 

IOO 

no 

1  20 

130 

tc 

11.7 

20.8 

32.6 

47.0 

64.0 

83.2 

1  06 

130-5 

158 

188 

221 

From  Table  XIX  it  becomes  evident  that  the  centrifugal 
tension,  which  diminishes  the  effective  tension-producing  pressure 
between  belt  and  pulley  upon  which  the  frictional  driving  power 


BELTS,  ROPES,  BRAKES,  AND  CHAINS.  301 

depends,  increases  rapidly  with  the  velocity  and  if  (1  =  240  Ibs. 
per  square  inch  there  will  be  no  effective  pressure  at  a  speed  of 
about  8000  ft.  per  minute.  In  other  words  the  belt,  if  put  on 
with  the  proper  value  of  /0  corresponding  to  /i  =  24o,  can  trans- 
mit no  power  at  this  speed,  because  the  centrifugal  force  is  so 
great  that  no  pressure  exists  between  the  belt  and  the  face  of 
the  pulley,  and  hence  there  is  no  friction. 

The  necessary  value  of  P  is  a  given  condition  in  any  problem. 
If  the  power  to  be  transmitted  by  the  belt  is  given  in  HP  and 
the  velocity  of  the  belt,  F,  in  feet  per  minute  is  known, 


The  most  economical  speed  at  which  to  use  a  leather  belt  is 
about  4500  ft.  per  minute.  In  general  P  is  determined  by 
dividing  the  foot-pounds  of  work  per  minute  to  be  transmitted, 
by  the  belt  speed  (or  pulley  rim  velocity)  in  feet  per  minute. 

The  value  of  6  is  determined  for  the  pulley  of  smallest  arc 
of  contact  from  the  diameters  of  the  pulleys  and  the  distance 
between  their  centers.  (See  sec.  174.) 

The  value  of  /*,  the  coefficient  of  friction,  varies  with  the 
kind  of  belting,  the  material  and  character  of  surface  of  pulley, 
the  condition  of  the  belt  as  regards  dressing,  the  side  of  the 
leather  used,  and  particularly  with  the  rate  of  slip  of  the  belt 
on  the  pulley.  This  slip  is  a  compound  of  two  factors,  actual 
slippage  and  belt  creep,  the  latter  being  the  unavoidable  move- 
ment of  the  belt  on  the  pulley  due  to  its  elasticity  and  the  dif- 
ference in  tension  between  the  tight  and  slack  sides.  Leather 
belting  is  extremely  variable  in  its  properties.  The  coefficient 
of  friction  for  oak-tanned  leather,  hair  side  on  a  cast-iron  turned 
pulley,  ranges  approximately  as  follows  :* 

*  Prof.  Lanza,  Trans.  A.  S.  M.  E.,  Vol.  VII.  See  also  paper  by  Wilfred  Lewis, 
same  volume  and  one  by  Prof.  W.  W.  Bird  in  Vol.  XXVI.  Another  good  paper 
is  that  of  F.  W.  Taylor,  Vol.  XV. 


302 


MACHINE  DESIGN. 


M 

Av.  vel.  of  slip  on 
pulley,  ft.  minute 

one 
=  Vs 

.12  tO   .17 

O 

.24 

2.  I 

.28 

2.6 

•31 

6.9 

•33 

IS 

.82 

210 

This  corresponds  roughly  to 


Prof.  Lanza  recommends  a  uniform  value  of  ^=.27.  This 
corresponds  to  a  uniform  slip  of  between  two  and  three  feet 
at  all  speeds. 

In  his  valuable  paper  in  Trans.  A.  S.  M.  E.,  Vol.  XXXI, 
Carl  Earth  writes  this  formula,  based  upon  his  own  experiments 
and  those  of  Prof.  Bird, 

2 


and  he  also  recommends, 


•54- 


140 


500+7* 


Equating  these  two  expressions  for  p, 

160+0.887 
s~  85  +0.037* 

These  formulae  are  open  to  criticism,  but  may  be  accepted 
tentatively  until  further  data  are  made  available. 

The   total  slip  on  two  pulleys  =  2  7S,    /.    per  cent   of   total 
200  Vs 


slip  = 


7 


174.  Efficiency  of  Belt  Drive. — An  approximate  estimate 
of  the  efficiency  of  a  belt  drive  can  be  made.  The  subscripts 
i  and  2  are  used  for  driver  and  follower,  respectively.  There 


BELTS,  ROPES,  BRAKES,  AND   CHAINS.  303 

are  three  losses  to  be  considered:  (a)  journal  friction,  (b)  work 
of  bending  the  belt,  (c)  slip  and  creep  friction.  Each  of  these 
losses  takes  place  at  each  pulley. 

(a)  The  journal  friction  work  per  minute  at  both  bearings, 
in  foot-pounds 

=  //(  (Ti  —  Tc)  +  (T2  —  Tc)  )(  -         h-         —I, 

\  /  \      12  12      / 

where  /x' =  coefficient  of  journal  friction  appropriate  to 

conditions  (see  Chaps.  XII  and  XIII) ; 
di  and  d2  =  journal  diameters,  inches; 
NI  and  A7"2  =  revs.  per  minute; 
TI,  T2,  and  Tc,  total  belt  tensions  as  above,  pounds. 

(b)  For  the  work  lost  in  bending  the  belt  the  following  formula, 
based    upon    Eytelwein's  for  ropes,  may  be  used  in  default  of 
one  based  upon  specific  investigation. 

Work  in  foot-pounds  per  minute  bending  belt 


.o48PF  -+— 
vi     r2 

P=belt  pull=Ti  -T2,  in  Ibs.; 
F  =  belt  velocity,  feet  per  minute; 
h  =  belt  thickness,  inches ; 
TI  and  r2  =  pulley  radii,  inches. 

(c)  The  friction  loss  due  to  belt  slippage  at  each  pulley  in 
foot-pounds  per  minute 


/£  =  coefficient  of  friction  of  belt  on  pulley  at  the  selected 

rate  of  slip; 
p  =  pressure  between  belt  and  pulley  face  per  linear  inch, 

pounds; 

r  =  radius  of  pulley,  inches; 
0  =  arc  of  contact,  radians ; 
s  =  slip  at  each  pulley,  feet  per  minute. 


304  MACHINE    DESIGN. 

This  transforms  into : 

Total  friction  work  at  both  pulleys  due  to  slippage  in  foot- 
pounds per  minute 


Summing  these  three  losses  (a),  (b),  and  (c)  in  foot-pounds 
per  minute,  subtracting  them  from  PV  and  dividing  the  result 
by  PV  will  give  the  efficiency  of  the  drive. 

175.  Problem. — A  single-acting  pump  has  a  plunger  8  inches 
=  0.667  foot  in  diameter,  whose  stroke  has  a  constant  length  of 
10  inches  =0.833  foot.  The  number  of  strokes  per  minute  is  50. 
The  plunger  is  actuated  by  a  crank,  and  the  crank-shaft  is 
connected  by  spur-gears  to  a  pulley-shaft,  the  ratio  of  gears 
being  such  that  the  pulley-shaft  runs  300  revolutions  per  minute. 
The  pulley  which  receives  the  power  from  the  line-shaft  is  18 
inches  in  diameter.  The  pressure  in  the  delivery-pipe  is  100  Ibs. 
per  square  inch.  The  line-shaft  runs  150  revolutions  per 
minute,  and  its  axis  is  at  a  distance  of  12  feet  from  the  axis  of 
the  pulley-shaft. 

Since  the  line-shaft  runs  half  as  fast  as  the  pulley- shaft,  the 
diameter  of  the  pulley  on  the  line-shaft  must  be  twice  as  great 
as  that  on  the  pulley-shaft,  or  36  inches.  The  work  to  be  done 
per  minute,  neglecting  the  friction  in  the  machine,  is  equal  to 
the  number  of  pounds  of  water  pumped  per  minute  multiplied 
by  the  head  in  feet  against  which  it  is  pumped.  The  number 
of  cubic  feet  of  water  per  minute,  neglecting  "slip,"  equals  the 
displacement  of  the  plunger  in  cubic  feet  multiplied  by  the 

o.66y2X7r 

number  of   strokes  per  minute  =—          — Xo. 833X50  =  14. 5 5, 

4 

and  therefore  the  number  of  pounds  of  water  pumped  per  minute 
=  14.55X62.4  =  908.  One  foot  vertical  height  or  "head"  of 
water  corresponds  to  a  pressure  of  0.433  ft>.  per  square  inch, 
and  therefore  100  Ibs.  per  square  inch  corresponds  to  a  "head" 


BELTS,  ROPES,  BRAKES,  AND  CHAINS. 


305 


of  100-^0.433  =  231  feet.  The  work  done  per  minute  in  pump- 
ing ths  water  therefore  is  equal  to  908  Ibs.  X23i  feet  =209,748 
ft.-lbs.  The  velocity  of  the  rim  of  the  belt-pulley  is  equal  to 
300x1.5X7:=  1 41 4  feet  per  minute.*  Therefore  the  force  P  =  T\ 
—  2^2  =  209,748  ft.-lbs.  per  minute  -^1414  feet  per  minute  = 
148  Ibs. 


To    find    a    (see     Fig.     182)     sin/?  = 


Therefore    ?  = 


144' 


=0.0625. 
5o';    a 


in  TT  measure  =  172! X 0.0175  =3.025  =6. 

TI 
log  —  =  0.4343  X  fJiXd  =0.4343  Xo.3  X3.025  =0.3941. 

J-  2 

rp 

•'•  -^-  =  2.48;  P  =  TI  —  7^2  =  14^* 

^  2 

Combining  these  equations  TI  is  found  to  be  equal  to  248  Ibs., 
the  maximum  stress  in  the  belt. 

/y-i  Q 

The  cross-sectional  area  of  belt  should  be  equal  to  —  =  — 

t\      240 

=  1.03  square  inch. 

Single-thickness  belting  varies  from  0.2  to  0.25  of  an  inch  in 
thickness,  hence  the  width  called  for  by  our  problem  would  be 
1.03 


0.2 


=  5  inches,  say. 


176.  Problem.  —  A  sixty-horse-power  dynamo  is  to  run  1500 
revolutions  per  minute  and  has  a  1  5-inch  pulley  on  its  shaft. 


At  this  speed  the  simple  form  of  the  belt  formula  may  be  used. 


306  MACHINE    DESIGN. 

Power  is  supplied  by  a  line-shaft   running  150  revolutions  per 
minute.     A  suitable  belt  connection  is  to  be  designed. 

The  ratio  of  angular  velocities  of  dynamo-shaft  to  line-shaft 
is  10  to  i  ;  hence  the  diameter  of  the  pulley  on  the  line-shaft 
would  have  to  be  ten  times  as  great  as  that  of  the  one  on  the 
dynamo,  =12.5  feet,  if  the  connections  were  direct.  This  is 
inadmissible,  and  therefore  the  increase  in  speed  must  be  ob- 
tained by  means  of  an  intermediate  or  counter  shaft.  Suppose 
that  the  diameter  of  the  largest  pulley  that  can  be  used  on  the 
counter-shaft  =48  inches.  Then  the  necessary  speed  of  the 

counter-shaft  =  i5ooX~^=:47o,  nearly.     The  ratio  of  diameters 
40 

of  the  required  pulleys  for  connecting  the   line-shaft  and  the 

47° 
counter-shaft  =•  —  -  =  3.13.     Suppose  that  a  6o-inch  pulley  can 

be  used  on  the  line-shaft,  then  the   diameter  of   the  required 

60 
pulley  for  the  counter-shaft  will  =  —  -  =  19  inches,  nearly.    Con- 

O"     O 

sider  first  the  belt  to  connect  the  dynamo  to  the  counter-shaft. 
The  work  =  60X33,000  =  1,980,000  ft.-lbs.  per  minute;   the  rim 

of  the  dynamo-pulley  moves  —  -X  1500  =  5890  feet  per  minute. 

Therefore  Ti-T2=  ^     -  =  336  Ibs.     The  axis  of  the  counter- 
5090 

shaft  is  10  feet  from  the  axis  of  the  dynamo,  and,  as  before, 


Therefore        £=7°  54'. 

a  =  180°  -20  =164°  12', 
6  =  164°.  2  Xo.oi75  =  2.874. 

The  nearest  value  of  F,  in  Table  XIX,  to  5890  is  6000,  and 
the  corresponding  value  of  tc  is  130.5. 
More  accurately 

4=126, 


BELTS,  ROPES,  BRAKES,  AND  CHAINS.  307 

140 

"=-54-S- 


—  1  2  O 

=  0.6442. 


fa-  152, 

-/2  =  240  -152  =  88, 


-  — 
n  —fa 


=        =  3-82  square  inches. 


A  double  belt  is  about  f  inch  thick.     Our  problem,  then, 
calls  for  a  double  belt  1X3.82  =  10  inches,  say,  wide. 

177.  Variation  of  Driving  Capacity.—  From  equation   (30'), 

T 
sec.  173,  it  follows  that  the  ratio  of  tensions,  —  r,  when  the  belt 

*2 

slips  at  a  certain  allowable  rate  (i.e.,  when  p  is  constant),  de- 
pends only  upon  a.  The  velocity  of  the  belt  also  remains 
constant.  This  ratio,  therefore,  is  independent  of  the  initial 

T 

tension,  TQ\    hence  "  taking  up"   a  belt  does  not  change  —  . 

•L  2 

The  difference  of  tension,  TI  -T2  =  P,  is,  however,  dependent 
on  TQ.  Because  p,  the  normal  pressure  between  belt  and 
pulley,  varies  directly  as  jT0,  then,  since  dF=fipds  =  dT,  it 
follows  that  dT  varies  with  7^  and  hence 


fdT=Tl-Ts 


varies    with    T0.     This    is    equivalent    to    saying   that    "  taking 
up  "  a  belt  increases  its  driving  capacity. 

This  result  is  modified  because  another  variable  enters  the 
problem.  If  TQ  is  changed,  the  amount  of  slipping  changes, 
and  the  coefficient  of  friction  varies  directly  with  the  amount 
of  slipping.  Therefore  an  increase  of  T0  would  increase  p 


308  MACHINE  DESIGN. 

and  decrease  //  in  the  expression  npds=dT,  and   the  converse 
is  also  true.     This  is  probably  of  no  practical  importance. 

The  value  of  P  may  also  be  increased  by  increasing  either  /*, 
the  coefficient  of  friction,  or  0,  the  arc  of  contact,  since  increase 

/T-i 

of    either    increases    the    ratio  — .    and     therefore    increases 

^2 

T1-T2=P. 

Increasing  TQ  decreases  the  life  of  the  belt.  It  also  in- 
creases the  pressure  on  the  bearings  in  which  the  pulley-shaft 
runs,  and  therefore  increases  frictional  resistance;  hence  a 
greater  amount  of  the  energy  supplied  is  converted  into  heat  and 
lost  to  any  useful  purpose.  But  if  T0  is  kept  constant,  and  /£ 
or  6  is  increased,  the  driving  power  is  increased  without  noticeable 
change  of  pressure  in  the  bearings,  since  \/7\  +  Vr2  remains 
constant.  When  possible,  therefore,  it  is  preferable  to  increase 
P  by  increase  of  //  or  6,  rather  than  by  increase  of  T0. 

Application  of  belt-dressing  may  serve  sometimes  to  in- 
crease fJL. 

If,  as  in  Fig.  179,  the  arrangement  is  such  that  the  upper 
side  of  the  belt  is  the  slack  side,  the  "sag  "  of  the  belt  tends  to 

T 
increase  the  arc  of  contact,  and  therefore  to  increase  ~.      If 

1  2 

the  lower  side  is  the  slack  side,  the  belt  sags  away  from  the  pulleys 

T" 

and  6  and  -^r  are  decreased. 

12 


FIG    183. 

An  idler-pulley,  C,  may  be  used,  as  in  Fig.  183.     It  is  pressed 
against  the  belt  by  some  means.     Its  purpose  may  be  to  increase 


(x/7"1  4-  \/y  \ 
_  I  -  -I2.     In  this  case 
2          / 


BELTS,  ROPES,  CHAINS,  AND   BRAKES.  309 

friction  in  the  bearings  is  increased,  and  this  method  should  be 

avoided.     Or  it  may  be  used  on  a  slack  belt  to  increase  the 

T 
angle  of  contact,  a,  the  ratio  -=r,  and  therefore  P,  the  driving 


* 

(\/  T*   _I_  \/  T1  \  2 
—  2  )  ,  may  be 
2  / 

made  as  small  a  value  as  is  consistent  with  driving,  and  hence 
the  journal  friction  may  be  small. 

Tighteners  are  sometimes  used  with  slack  belts  for  dis- 
engaging gear,  the  driving-pulley  being  vertically  below  the 
follower. 

In  the  use  of  any  device  to  increase  fj.  and  a,  it  should  be 
remembered  that  T\  is  thereby  increased,  and  may  become 
greater  than  the  value  for  which  the  belt  was  designed.  This 
may  result  in  injury  to  the  belt. 

In  Fig.  184,  the  smaller  pulley,  A,  is  above  the  larger  one, 
B.  A  has  a  smaller  arc  of  contact,  and  hence  the  belt  would 


FIG.  184.  FIG.  185. 

slip  upon  it  sooner  than  upon  B.  The  weight  of  the  belt,  how- 
ever, tends  to  increase  the  pressure  between  the  belt  and  A9 
and  to  decrease  the  pressure  between  the  belt  and  B.  The 
driving  capacity  of  A  is  thereby  increased,  while  that  of  B  is 
diminished;  or,  in  other  words,  the  weight  of  the  belt  tends  to 
equalize  the  inequality  of  driving  power.  If  the  larger  pulley 
had  been  above,  there  would  have  been  a  tendency  for  the 
belt  weight  to  increase  the  inequality  of  driving  capacity  of  the 


310  MACHINE  DESIGN. 

pulleys.     The  conclusion  from  this,  as  to  arrangement  of  pulleys, 
is  obvious.     For  effect  of  humidity  see  Jour.  A.S.M.E.,  1915. 

178.  Proper  Size   of  Pulleys. — A  belt  resists  a  force  which 
tends  to  bend  it.     Work  must  be  done,  therefore,  in  bending  a 
belt  around  a  pulley.     The  more  it  is  bent  the  more  work  is 
required  and  the  more  rapidly  the  belt  is  worn  out.     Suppose 
AB,  Fig.  185,  to  represent  a  belt  which  moves  from  A  toward 
B.      If  it  runs  upon  C  it    must  be  bent  more  than  if  it  runs 
upon  D.     The  work  done  in  bending  the  belt  is  converted  into 
useless  heat  by  the  friction  between  the  belt  fibers.    It  is  desirable, 
therefore,  to  do  as  little  bending  as  possible.    This  is  one  reason 
why  large  pulleys  in  general  are  more  efficient  than  little  ones. 
The  resistance  to  bending  increases  with  the  thickness  of  the 
belt,  and  hence  double  belts  should  not  be  used  on  small  pulleys 
if  it  can  be  avoided. 

Double        belts  may  be  used  on  pulleys  12"  and  over. 
Triple  "       "     "      "     "        "      20"    " 

Quadruple     "       "     "      "     "        "      30"    "       " 

179.  Distance  Desirable  between   Shafts. — In  the  design  of 
belting  care  should  be  taken  not  to  make  the  distance  between 
the  shafts  carrying  the  pulleys  too  small,  especially  if  there  is  the 
possibility  of  sudden  changes  of  load.     Belts  have  some  elasticity, 
and  the  total  yielding  under  any  given  stress  is  proportional  to 
the  length,  the  area  of  cross-section  being  the  same.     There- 
fore a  long  belt  becomes  a  yielding  part,  or  spring,  and  its  yielding 
may  reduce  the  stress  due  to  a  suddenly  applied  load  to  a  safe 
value;  whereas  in  the  case  of  a  short  belt,  with  other  conditions 
exactly  the  same,  the  stress  due  to  much  less  yielding  might  be 
sufficient  to  rupture  or  weaken  the  joint. 

180.  Rope-drives. — The  formulae  which  have  been  derived 
for  belts  also  apply  to  rope-drives.      For  good  durability  the 
allowable  tension  in  a  rope-drive  should   be  about  200^  Ibs. 
where  d  is  the  diameter  of  the  rope  in  inches.     Experiments 


BELTS,  ROPES,  CHAINS,  AND    BRAKES. 


vary  greatly  in  the  value  of  the  coefficient  of  friction  for  a  well- 
lubricated  rope  on  a  flat-surfaced  smooth  metal  pulley.  It  may 
be  taken  equal  to  0.12.*  But  ropes  are  not  commonly  used 
on  flat  pulleys;  instead  of  this  they  run  in  grooves  on  the  faces 
of  sheave  wheels,  and  substitution  must  be  made  for  //  in  the 

angle  of  groove 
formula  (3),  not  0.12  but  o.i2Xcosec  -  — -. 

The  following  table  gives  the  values  of  /*  for  different  angles 

of  grooves. 

TABLE  XX. 


Angle  of  groove  in  degrees  .  .  . 
n  .  .                                   

3° 
0.46 

35 
0.40 

40 

O.  7"J 

45 
o.  31 

5° 
0.28 

55 
0.26 

60 

o  24. 

Fibrous  ropes  for  power  transmission  purposes  are  made 
chiefly  of  cotton  or  manila  fiber.  The  former  is  softer,  more 
flexible  and  elastic,  the  latter  is  cheaper  and  stronger.  The 
former  weighs  about  10  per  cent  less  than  the  latter.  The  fol- 
lowing tables  are  computed  for  manila  rope. 

Taking  the  weight  of  rope  per  linear  inch  =  0.0268^2  Ibs., 
and  the  allowable  tension,  T1  =  2ood2  Ibs.,  and  solving  for  Te 
at  various  speeds,  gives  the  following  results: 

TABLE  XXI. 


V 
Tc 

d* 

IOOO 

2.78 

2OOO 
II.  I 

2500 
17.4 

3000 
25.0 

3500 

34-0 

4000 
44-4 

4500 
56.2 

5000 
69.5 

5500 
84.0 

6000 

IOO 

6500 
118 

7000 
136 

7500 
156 

8000 
178 

8500 
200 

V  is  the  velocity  of  the  rope  in  feet  per  minute. 
For  convenience  the  following  table  is  given  showing  the  cor- 
responding values  of  angles  in  degrees  and  circular  measure: 

*  "Rope-driving,"  by  J.  J.  Flather.     New  York:  Wiley  &  Sons. 


312 


MACHINE  DESIGN. 


TABLE  XXII. 


i 

a.  .. 

105 

120 

135 

T5° 

165 

1  80 

195 

2IO 

240 

0.  .. 

1.83 

2.09 

2-35 

2.62 

2.88 

3-i4 

3-43 

3.66 

4.19 

It  will  be  remembered  that  #=0.01750:. 

The  diameter  of  the  sheave  wheel  is  properly  calculated  from 
the  point  of  tangency  of  the  rope  to  the  groove  (A-A,  Fig.  186) 
and  not  from  the  middle  of  the  rope  O.  The  diameter  of  the 
sheave  wheel  should  not  be  too  small  or  the  rope  will  wear  out 
very  rapidly.  The  following  table  gives  the  minimum  values 
D  being  the  diameter  of  the  wheel  and  d  that  of  the  rope : 

TABLE  XXIII 


d.. 

3" 

l" 

l|" 

if" 

i!" 

2" 

D 

24. 

l6 

48 

60 

72 

9.A 

Table  XXIII  is  for  general  purposes.  Specifically  the  formula 
D  =  d1'7X*V+i2  inches  may  be  used. 

Two  systems  of  rope-driving  are  in  use,  the  English  and 
the  American.  In  the  former  a  number  of  ropes  are  used  side 
by  side.  In  the  latter  a  single  continuous  rope  is  used  with 
a  guide  and  tightener.  As  long  as  all  the  grooves  in  each 
sheave  are  alike,  each  rope  will  tend  to  carry  its  proportionate 
share  of  the  load  in  the  English  system,  provided  all  the  ropes 
had  the  same  original  tension,  and  have  stretched  the  same 
amount. 

Next  to  the  angle  of  groove  the  most  important  item  is  to 
have  the  grooves  as  smoothly  surfaced  as  possible. 

Fig.  i&6A  shows  one  manufacturer's  standard  groove  pro- 
portions for  fibrous  ropes. 

The  diameters  of  multiple-grooved  sheaves  must  be  accurately 
alike. 

For  the  American  system  the  grooves  In  tne  larger  sheave 


BELTS,  ROPES,  BRAKES,  AND  CHAINS. 


313 


should  have  a  greater  angle  than  those  in  the  smaller  sheave,  or  the 
load  will  be  unequally  divided  among  the  various  wraps  of  the  rope.* 


-El 


At 


-So    «g 

II  I 


'    111    9 


o 


FIG.  i 86. 

The  tension  in  each  wrap  of  the  rope  will  be  the  same,  when 
running,  if  the  friction  on  each  sheave  wheel  is  the  same.  The 
friction  on  each  pulley  will  be  the  same  if  the  products  of  the 
arcs  of  contact  by  the  respective  coefficients  are  equal. 

Y 

Let  /£  =  coefficient  of  larger  sheave =0.12  cosec— ; 
•y  =  angle  of  groove  of  larger  sheave ; 
a  =  arc  of  contact  of  larger  sheave ; 

*  See  further  Proc.  Am.  Soc.  C.  E.,  Vol.  XXIII.  Mr.  Spencer  Miller  on 
Rope-driving. 


314 


MACHJNh   DbSIGN. 


fjf  =  coefficient  of  smaller  sheave  =0.12  cosec  — : 

2 

f  =  angle  of  groove  of  smaller  sheave; 
a!  =arc  of  contact  of  smaller  sheave. 
Then  //a  should  =  //a' '. 

r  f     of 

.'.  cosec  —  = cosec  —  X— . 
2  2      a 

The  following  table  gives  the  proper  values  for  equal  adhesion: 
TABLE  XXIV. — ANGLE  OF  GROOVE  FOR  EQUAL  ADHESION. 


Arc  of  contact  on  small  pulley     a.' 

0 

Arc  of  contact  on  large  pulley     a 
Angle  of  groove  in  large  pulley  when 
groove  in  small  pulley  =  35°  

0.9 
40° 

44° 

°-75 

47° 

0.7 

<a° 

0.05 
s<;0 

o.  6 
60° 

Angle  of  groove  in  large  pulley  when 
groove  in  small  pulley  =  40°  

4S° 

50° 

'U0 

<;8° 

64° 

70° 

Angle  of  groove  in  large  pulley  when 
groove  in  small  pulley  =  45° 

so0 

C<° 

60° 

66° 

72° 

80° 

00 

I* 

The  angle  of  groove  on  the  smaller  sheave  wheel  is  generally 
made  45°.  Assuming  this,  an  angle  of  contact  of  165°,  and  an 
allowable  stress  =  2Ood2,  the  following  table  has  been  com- 
puted for  the  horse-power  transmitted  by  each  wrap  of  the  ropj: 

TABLE  XXV. — HORSE  POWER  TRANSMITTED  BY  SINGLE  ROPE. 


Velocity 

•i  rope  in 

ft.  per 

Diameter  of  rope  in  inches. 

nun. 

V. 

i 

I 

i 

ii 

ii 

ii 

2 

1000 

1.38 

1.98 

3-52 

5-50 

7.91 

ii  .20 

14.08 

2000 

2.64 

3-8o 

6-75 

10.55 

15.20 

21.50 

27.00 

2500 

3-i8 

4.58 

8.15 

12.70 

18.35 

26.00 

32.60 

3000 

3-67 

5-28 

9.40 

14.70 

21.15 

30.00 

37.60 

3500 

4.07 

5.85 

10.40 

16.25 

23.40 

33-20 

41  .60 

40OO 

4-36 

6.28 

11.15 

17.40 

25.10 

35-6o 

44.60 

4500 

4-54 

6-54 

ii  .60 

18.12 

26.10 

37.00 

46.40 

5000 

4-57 

6-59 

ii  .70 

18.30 

26.35 

37-30 

46.80 

5500 

4-45 

6.42 

ii  .40 

17.80 

25-65 

36.40 

45.60 

6OOO 

4-2 

6.05 

10-75 

16.80 

24.20 

34-30 

43-oo 

6500 

3-73 

5-37 

9-55 

14-93 

21.50 

30.40 

38.20 

7OOO 

3.12 

4-50 

8.00 

12.50 

18.00 

25-50 

32.00 

7500 

2.31 

3-32 

5-9° 

9.21 

13-30 

18.80 

23.60 

BELTS,  ROPES,  BRAKES,  AND  CHAINS.  315 

For  durability  a  few  turns  of  a  larger  rope  are  preferable  to 
more  turns  of  a  smaller  rope. 

The  most  economical  speed,  taking  first  cost  and  relative  wear 
into  consideration,  is  about  4500  feet  per  minute. 

In  any  given  case,  since  Ti  =  2ood2  and  Tc  =  o.oiv2d2,  T2 
can  be  computed  by  writing  for  it  xd2.  Substituting  these  values 
in  equation  (3), 

200d2  -O.OIV2d2 

10S 


The  term  d2  divides  out,  x  can  be  solved  for,  and  the  value 
of  T2  determined  from  T2  =  xd2. 

The  initial  tension,  TQ,  is  determined  from  the  equation 


To=( 


The  length  and  deflections  of  the  rope  are  important  points. 
The  curve  the  rope  takes  between  supports  is  the  catenary  and 
the  following  equations  are  approximately  correct  for  horizontal 
drives:* 

C2w 


A  =  deflection  at  middle  of  span  in  feet; 

C=span,  in  feet; 

w  =  weight  of  rope  per  linear  foot; 


*  For  inclined  drives,  etc.,  see  Reuleaux's  Constructor  or  Flather's  Rope  Driving. 
Trade  catalogs  of  the  C.  W.  Hunt  Co.,  the  Plymouth  Cordage  Co.,  and  others, 
give  valuable  data  on  groove  forms,  arrangement  of  installations,  etc. 


316  MACHINE  DESIGN. 

T=  tension  in  pounds  (T0j  T\,  or  T2  for  40,  A\,  or  ^2); 
L  =  length  of  catenary  in  feet.     For  original  length  use  T0 
and    J0-     The    entire    rope    length    for    one   wrap  = 


RI,  R2  =  pulley  radii,  in  feet; 
0i,  0-2  =  corresponding  arcs  of  contact,  radians. 

181.  Efficiency  of  Rope  Drives.  —  The  claims  made  for  rope 
driving  embrace:  suitability  to  transmitting  large  amounts  of 
power,  quiet  running,  can  be  carried  in  any  direction,  can  be 
subdivided  most  readily,  does  not  require  accurate  alignment  of 
sheaves,  freedom  from  electrical  disturbance,  reasonably  weather 
proof,  economy  in  first  cost  and  in  maintenance.  E.  H.  Ahara,  in 
Trans.  A.  S.  M.  E.,  Vol.  XXXV,  reports  on  a  series  of  about  700 
tests,  extending  over  a  continuous  period  of  five  months.  His 
results  show  higher  efficiencies  for  the  American  system  than 
for  the  English  and  higher  efficiencies  for  the  open  drive  than 
the  "  up  and  over,"  with  either  system.  He  used  extreme  belt 
tensions,  in  some  cases  T2  =  ^6od2,  therefore  transmitting  as  much 
as  four  times  the  power  suggested  as  economical  in  Table  XXV. 
This  was  done  without  loss  in  efficiency,  whatever  may  have 
been  the  result  on  the  ultimate  life  of  the  rope.  Data  collected 
by  ].  ].  Flather  indicate  that  ropes  in  which  Tl  greatly  exceeds 
2ood2  wear  out  with  serious  rapidity.  The  Ahara  tests  show 
American  open  drives  under  best  conditions  averaging  about 
93  per  cent,  English  open  drives  87  per  cent,  American  "  up 
and  over  "  80  per  cent,  English  "  up  and  over,"  75  per  cent. 
They  seem  to  show  "  that  the  efficiency  in  rope  driving  is  con- 
siderably greater  at  the  lower  speeds  than  at  the  higher  ones, 
the  dropping  off  being  especially  noticeable  above  4500  ft.  per 
minute  of  rope  speed.  They  also  show  that  the  efficiency  of  a 
rope  drive  is  not  materially  affected  by  distances  between  centers 
up  to  150  ft.,  that  the  drop  of  efficiency  at  50  per  cent  load  is 
comparatively  small  over  that  of  full  load,  and  that,  if  proper 


BELTS,  ROPES,  BRAKES,  AND   CHAINS.  317 

care  is  exercised  to  have  all  grooves  perfect  in  pitch  diameter, 
many  as  well  as  few  ropes  can  be  run  on  a  drive  with  good 
efficiency." 

182.  Problem.  —  An  engine,  running  at  100  revs,  per  minute 
and  delivering  275  H.P.  is  to  drive  a  main-shaft  60  ft.  distant 
at  300  revs,  per  minute.  Conditions  permit  the  use  of  sheaves  of 
most  economical  size. 

Select  7  =  4500  ft.  per  minute. 

V 
Engine  sheave  diameter  —  --  =14.291  ft. 

.  7 
Shaft  sheave  diameter  =  —      —  =  4.  764  ft. 

o 

Allowable  rope  diameter,  from 


where 

D  =  smaller  sheave  diameter  in  inches,  ^  =  1.5  inches. 
Angle  of  contact  on  smaller  pulley  = 

180°  -2  sin'1  7'14    ~2'3  2  =  i7i°  =  3  radians. 
oo 

Angle  of  contact,  larger  pulley  =  3.  283  radians. 
Tc  from  Table  XXI  =  56.2^.    ^  =  0.31. 

200  -56.2 


Tl-T?  =  20od2  -  1  13^2  =  87^2. 
=  196  Ibs. 

4500X196 

H.P.  per  wrap-3^-        —=26.7. 
33,000 


MACHINE  DESIGN. 


Number  of  wraps  =  -^—=10.5,  say  10  wraps. 


Length  of  one  wrap  (under  tension  T0)  = 

2X60.04  +7.146X3-283  +2.382X3  =  150.7  ft. 

Length  of^ten  wraps  =1507  ft. 

Add  allowance  for  splice  and  tension  carriage. 

.322^2 


60^X03 


183.  Wire  Rope  Transmission.*  —  For  many  years  wire  rope 
has  been  used  satisfactorily  for  power  transmission.  It  is  not 
particularly  applicable  to  short  spans  (where  they  are  under 
60  ft.  it  is  not  possible  to  splice  the  rope  with  such  a  degree  of 
nicety  as  to  give  the  desired  tensions  and  deflections,  and  some 
mechanical  adjustment  becomes  necessary)  but  it  is  applicable 
to  long  spans.  At  Lockport,  N.  Y.,  a  clear  span  of  1700  ft., 
without  intervening  support,  has  been  used.  The  length  of  clear 
span  is  determined  by  the  allowable  deflections.  When  the 
distance  exceeds  the  limit  for  a  clear  span,  supporting  sheaves 

*  See  Reuleaux's  Constructor. 


BELTS,  ROPES,  CHAINS,  AND  BRAKES.  319 

with  unfilled  grooves  (idlers)  are  used.  In  very  long  trans- 
missions it  is  impracticable  to  run  the  rope  at  high  velocities. 
The  force  factor  must  therefore  be  made  greater,  and  this  is 
done  by  increasing  the  angle  of  contact  by  lapping  the  rope 
several  times  about  a  pair  of  grooved  drums  at  each  end.  The 
sheaves  used  should  be  rilled  with  hard  wood,  tarred  oakum, 
or  segments  of  leather  and  rubber  soaked  in  tar  and  packed 
alternately  in  the  groove.  The  sheaves  should  be  accurately 
balanced.  The  rope  rests  free  on  the  packing  and  does  not 
wedge  in  the  groove  as  with  fibrous  rope  transmission. 

The  same  general  formulae  apply  as  developed  for  fibrous 
ropes. 

Under  ordinary  conditions,  six-strand  ropes  of  seven  wires 
to  the  strand,  laid  about  a  hemp  core,  are  best  adapted  to  the 
transmission  of  power,  but  conditions  often  occur  where  twelve- 
wire  or  nineteen-wire  rope  is  to  be  preferred. 

The  weight  of  cast-steel  rope  per  linear  inch  is  approximately 

w=  — .     Its  ultimate  tensile  strength  is  approximately  64,000^2. 

For  plough  steel  this  is  y6,oood2  to  90,000^.  For  Swedish  iron, 
30,000^2. 

The  stress  induced  by  bending  the  rope  around  the  sheave 
is  computed  from  the  formula  (Bach), 


Ed 


/6=flexural  stress,  pounds  per  square  inch; 

E  =  modulus  of  elasticity,  29,000,000  for  steel; 

D  =  diameter  of  sheave,  inches; 

£  =  diameter  of  individual  wires,  inches 
=  i  diam.  of  rope  for  7 -wire  rope 
=-rV  diam.  of  rope  for  i9-wire  rope. 


320 


MACHINE  DESIGN. 


The  maximum  safe  tension  is  taken  at  one-fourth  the  ulti- 
mate strength  and  equals  about  1 6,000 d2  for  cast-steel  ropes. 
Three-fourths  of  this  (i2,oood2)  may  be  allowed  for  bending 
stress  and  one-fourth  (4000^)  for  working  tension.  On  this 
basis  Tables  XXVII-XXIX  have  been  developed. 


TABLE  XXVI.— VALUES  OF  n  FOR  WIRE  ROPE. 


Dry  rope  on  a  grooved  iron  drum 

Wet  rope  on  a  grooved  iron  drum 

Greasy  rope  on  a  grooved  iron  drum 

Dry  rope  on  wood-filled  sheaves 

Wet  rope  on  wood-filled  sheaves , 

Greasy  rope  on  wood-filled  sheaves 

Dry  rope  on  rubber  and  leather  filling. .  . , 
Wet  rope  on  rubber  and  leather  filling 
Greasy  rope  on  rubber  and  leather  filling , 


.120 
.085 
.070 

•235 
.170 
.140 

•495 
.400 
.205 


TABLE  XXVII. — DIAMETERS  OF  MINIMUM  SHEAVES  IN  INCHES. 


Diam.  of  Rope. 

Steel. 

Iron. 

7-Wire. 

ig-Wire. 

7-Wire. 

1  9-  Wire. 

24 

14 

48 

28 

30 

17 

60 

34 

| 

36 

21 

72 

42 

A 

41 

24 

83 

48 

| 

47 

28 

94 

56 

A 

53 

31 

1  06 

62 

t 

59 

35 

118 

70 

f 

65 

70 

38 
42 

130 
140 

76 
84 

i 

82 

49 

164 

98 

1 

94 

56 

188 

112 

TABLE  XXVIII.— DEFLECTION  OF  WIRE  ROPES. 

Steel  Iron 

Def.  of  still  rope  at  center  in  feet,         A0=  .000069  C2  AO=  .000138  C2 

Def.  of  driving  rope  at  center  in  feet,    Ai=. 000049  Cz  Ai=. 000098  C2 

Def.  of  slack  rope  at  center  in  feet,        A2=  .000103  C2  A2=  .000206  C2 
J*=span,  in  feet. 


BELTS,  ROPES,  BRAKES,  AND   CHAINS.  321 

TABLE  XXIX. — HORSE-POWERS  FOR  A  STEEL  ROPE  MAKING  A  SINGLE  LAP  ON 
DRY,  WOOD-FILLED,  MINIMUM  SHEAVES. 


Diameter 
of  Rope  in 
inches. 

Velocity  of  Rope  in  Ft.  per  Second. 

10 

20 

30 

40 

So 

60 

l 

2-5 

4-75 

7 

9-5 

"•5 

13-5 

A 

3-75 

7-5 

ii 

14-5 

18 

21.5 

1 

5-5 

10.5 

16 

21 

26 

30-5 

T^ 

7-5 

U-5 

21-5 

28.5 

35-5 

42 

i 

9-5 

28 

37 

46 

54-5 

TS 

12 

24 

36 

47 

58.5 

69 

| 

15 

30 

44 

58 

72 

85 

il 

18 

36 

53-5 

71 

87-5 

103 

| 

21.5 

43 

63.5 

84 

104 

124 

1 

29 

58 

86.5 

114 

141 

167 

I 

38 

76 

H3 

149 

185 

218 

The  horse-power  that  may  be  transmitted  by  iron  ropes  is 
one -half  of  the  above. 

Table  XXX  (p.  322)  gives  one  manufacturer's  standard 
sheave  wheel  proportions  for  steel  and  iron  ropes. 

184.  Steel  Belting.* — The  use  of  flat  steel  belts,  ranging 
from  0.008  to  0.03  inch  in  thickness,  in  place  of  leather  belts 
has  been  a  development  of  recent  years.  The  advantages  claimed 
for  them  are : 

(a)  Steel  belts  need  be  only  from  one-half  to  one-fourth  the 
width  of  leather  belts.     This  means  pulleys   of  narrower  face 
and  less  weight  and  cost. 

(b)  Their  own  first  cost  is  considerably  below  that  of  leather 
or  rubber  belting. 

(c)  They  do  not  stretch  or  slip  after  being  put  on  the  pulleys 
properly. 

(d)  They  are  not  appreciably  affected  by  variations  in  tem- 
perature or  moisture.     This  makes  them  very  reliable  for  use 
in  damp  places.     They  are  especially  adapted  for  use  in  paint 


*  American  Machinist,  Vol.  37,  pp.  852-3. 


322 


MACHINE  DESIGN. 


ooooooooooooocooooooootowi 


§O  O  "1  O  O  »/V  M  iO\O  O  00  f)  i/500  M 
O  ON  t-  O  VO  J-  O  O\O  •*  O  00  O  O  O\ 
•^•oo  fO  O\00  ^  N  M  ro  M  M  M 


2  rt"^ 


£ 


-W.  O- 

M   N   O\0000   t^O     I     ' 


t^oo  O  O  t^vO 


M  M  *t  t*3  f*3  CO 


BELTS,  ROPES,  BRAKES,  AND   CHAINS. 


323 


or  varnish  works,  etc.,  as  they  can  be  washed  off  readily  with 
gasoline. 

On  the  other  hand,  they  are  more  sensitive  than  other  belts, 
and  shafts  and  pulleys  must  be  in  line  and  level  or  the  belt  will 
run  to  the  low  side  of  the  pulley  and  may  run  off.  Crown  pulleys 
cannot  be  used  for  them  in  any  case.  They  will  run  on  flat- 
faced,  uncovered,  iron  or  steel  pulleys,  as  well  as  on  wood  pulleys, 
but  the  use  of  canvas  or  rubber  pulley  covering  is  so  beneficial 
that  it  seems  almost  necessary  to  good  service. 

185.  Block  and  Band  Brakes. — For  the  sake  of  unity  of 
treatment  block  brakes  will  be  considered  at  this  point.  The 
formulae  are  all  developed  by  equating  to  zero  the  sum  of  the 
moments  about  the  axis  of  the  brake  lever,  considering  the  latter 
as  a  free  body. 

F  =  force  in  pounds  at  end  of  brake  lever; 

P  =  tangential  force  in  pounds  at  rim  of  brake  wheel,  called 
braking  force ; 

/*  =  coefficient  of  friction  between  brake  block  and  brake  wheel. 

i.  Block   brake,  Fig.  iS6B.     For  rotation  in 
either  direction : 

Pb 


F  = 


M 

\u/' 


2.  Block  brake,  Fig.  i86C.  Upper  sign 
for  clockwise,  lower  sign  for  counter-clockwise 
rotation : 

Pb  /i 


FIG.  1 865' 


F  = 


a  +  b\fji 

3.  Block  brake,  Fig.  iS6D.  Upper  sign 
for  clockwise,  lower  sign  for  counter-clockwise 
rotation : 

p^ILti. 

a  +  b\u.~ 


FIG.  i86C. 


I.. 


FIG.  i86D. 


MACHINE  DESIGN. 


The  brake  wheel  and  friction-block  may  be  grooved  as  shown 
in  Fig.    i86E.      In  this   case  substitute   for  /*  in  the  foregoing 


equations  the  value 


-,  where  a  is 


FIG.  i86E. 


sma+//  cos  a 

one-half  the  angle  included  by  the  faces  of  the 
grooves. 

The  formulas  for  band  brakes,  simple  and 
differential,   are  developed  from  the  belt  for- 

TI  Tl 

mula,  loge  —  =/j.6,  which  may  be  written  —  =  ef  ,  e  being  the  base 
1%  1 2 

of  natural  logarithms  =  2. 7 182 8.      6  is  the  angle  of  contact  of 
brake  and  drum  in  radians. 

p  is  the  coefficient  of  friction. 

P  =  tangential  force  in  pounds  at  rim  of  brake  drum; 

F  =  force  in  pounds  at  end  of  brake  lever; 

TI  =  tension  in  tight  side  of  band,  pounds; 

T2  =  tension  in  slack  side  of  band,  pounds. 


1.  Simple  band  brake,  Fig.  i86F. 

Pbi   e*0    \ 

For  clockwise  rotation:  F  =  — (  —-» ). 

a  V*-i/ 

For  counter-clockwise  rotation:  F  =  — (    e  _    ). 

2.  Simple  band  brake,  Fig.  i86G. 

For  clockwise  rotation :  F  = — (  — ) . 

Pbl   <**    \  ^'^ « > 

For  counter-clockwise  rotation :  F = — I  — -Q )  •  ^ 

a  \er  - 1/  FlG  l86G 


BELTS,  ROPES,  BRAKES,  AND   CHAINS.  325 

3.  Differential  band  brake,  Fig.  iS6H. 

p/b2e"B  -bi\ 
For  clockwise  rotation :  F  =  —  {  — -^ —  -  ) . 

a\  e"9  -i   I 

P 

For  counter-clockwise  rotation:  F  =  — 


In  this  case  if  b2  is  equal  to  or  less  than  b^9,  F  will  be  zero 
or  negative  and  the  brake  sets  automatically. 


FIG.  i86ff .  FIG.  1 867. 

4.  Differential  band  brake,  Fig.  i86/. 
For  clockwise  rotation:  F  = 


P/b2eflB  +  bl\ 
ion  :  F  =  —  {  —  -a  -  )  . 
a\  4*  -i  / 


For  counter-clockwise  rotation  :  F  =  —  {  -^  -  ~  )  . 

a\  e^  -i   / 


If  b1=b2  =  b,  for  both  cases:  F  =— 


. 
a  \er  -  1/ 

1  86.  Chains.  —  Various  types  of  chains  are  used  for  power 
transmission  purposes. 

(a)  Tests  made  at  the  University  of  Illinois  *  show  that 
the  ordinary  method  of  computing  round-rod  chain  strength, 
both  open  and  stud  link,  is  incorrect.  They  show  that  a  load 
P  on  the  link,  while  it  produces  an  average  intensity  of  stress 

p 

in  the  cross-section  containing  the  minor  axis  =  —  ,  a  being  the 

2d 
*  Bulletin  No.  18. 


326 


MACHINE  DESIGN. 


area  of  the  rod  =  — ,  produces  a  much  greater  maximum  fiber 
4 

stress  than  this.  With  an  open  link  of  usual  proportions  the 
maximum  tensile  stress  is  approximately  four  times  this  value, 

2p 

or  — .     The  introduction  of  a  stud  in  the  link  equalizes  the 
a 

stresses  somewhat  throughout  the  link  and  reduces  the  maximum 
tensile  stress  about  20  per  cent.  The  following  formulae  are 
applicable  to  chains  of  the  usual  form: 

P  =  o.4^2/f,  for  open  links; 
p  =  0.5</%  for  stud  links; 

ft  =  allowable    unit    tensile    stress  =  15,000    to    20,000    (maxi- 
mum) pounds  per  square  inch. 


1 

1 

| 

FIG.  1  867. 

FIG.  i&6K. 

(b)  Flat  link  chains  of  either  the  block  or  roller  type,  Figs. 
1 867  and  K,  are  used  for  power  transmission  purposes.  They 
are  not  suitable  for  high  speeds.  Not  only  do  they  tend  to  wear 
and  stretch,  which  throws  them  out  of  equality  of  pitch  with  their 
sprockets,  but  from  the  nature  of  their  construction  it  is  inev- 
itable that  they  transmit  motion  at  a  continuously  varying  velocity 
ratio.  They  have  a  wide  field  of  usefulness,  however,  for  low- 
speed  transmission  where  an  absolutely  uniform  velocity  ratio 
is  not  required  nor  noise  objectionable.  Their  proportioning 
involves  two  chief  points,  the  shearing  strength  of  the  pins  and 
the  tensile  strength  of  the  side  plates  at  their  minimum  section. 
If  kept  free  from  dust,  well-lubricated,  not  overloaded  and  not 
run  at  too  high  a  speed,  they  show  efficiencies  ranging  as  high  as 
94  per  cent. 


BELTS,  ROPES,  BRAKES,  AND  CHAINS. 


327 


(c)  Some  of  the  objections  to  flat  link  chains,  particularly 
those  connected  with  change  of  pitch  of  chain  and  sprocket 
with  stretch  and  wear,  have  been  met  and  conquered  in  so-called 
silent,  high-speed  chains,  such  as  the  Renold  and  Morse.  These, 
because  they  continue  to  fit  their  sprockets,  run  with  much  less 


FIG.  i86L. 

noise  than  block  or  roller  chains,  may  be  run  at  considerably 
higher  speeds,  will  transmit  much  greater  powers,  and  when 
properly  installed  and  cared  for  show  extremely  high  efficiencies. 

Fig.  1 861,,  from  the  Morse  Chain  Company's  catalog,  shows 
the  action  of  this  chain  and  clearly  illustrates  their  employment 
of  the  extremely  efficient  knife-edge  bearing  at  the  joints. 


CHAPTER  XVI. 

FLY-WHEELS   AND    PULLEYS. 

187.  Theory  of  Fly-wheel.  —  Often  in  machines  there  is 
capacity  for  uniform  effort,  but  the  resistance  fluctuates.  In 
other  cases  a  fluctuating  effort  is  applied  to  overcome  a  uniform 
resistance,  and  yet  in  both  cases  a  more  or  less  uniform  rate 
of  motion  must  be  maintained.  When  this  occurs,  as  has  been 
explained,*  a  moving  body  of  considerable  weight  is  interposed 
between  effort  and  resistance,  which,  because  of  its  weight, 
absorbs  and  stores  up  energy  with  increase  of  velocity  when 
the  effort  is  in  excess,  and  gives  it  out  with  decrease  of  velocity 
when  the  resistance  is  in  excess.  This  moving  body  is  usually 
a  rotating  body  called  a  fly-wheel. 

To  fulfill  its  office  a  fly-wheel  must  have  a  variation  of 
velocity,  because  it  is  by  reason  of  this  variation  that  it  is  able 
to  store  and  give  out  energy.  The  kinetic  energy,  £,  of  a 
body  whose  weight  is  W  Ibs.,  moving  with-  a  velocity  v  feet  per 
second,  is  expressed  by  the  equation 

Wv2 

To  change  E,  with  W  constant,  v  must  vary.  The  allowable 
variation  of  velocity  depends  upon  the  work  to  be  accomplished. 
Thus  the  variation  in  an  engine  running  electric  lights  or  spin- 
ning-machinery should  be  very  small,  probably  not  greater 

*  See  §  43- 

328 


FLY-WHEELS  AND  PULLEYS.  329 

than  a  half  of  one  per  cent,  while  a  pump  or  a  punching- 
machine  may  have  a  much  greater  variation  without  interfering 
with  the  desired  result.  If  the  maximum  velocity,  v\,  of  the 
fly-wheel  rim  and  the  allowable  variation  are  known,  the  mini- 
mum velocity,  ^2,  becomes  known;  and  the  energy  that  can  be 
stored  and  given  out  with  the  allowable  change  of  velocity  is 
equal  to  the  difference  of  kinetic  energy  at  the  two  velocities. 


2g  2g          2g 

188.  The  general  method  for  fly-wheel  design  is  as  follows: 
Find  the  maximum  energy  due  to  excess  or  deficiency  of  effort 
during  a  cycle  of  action,  =  AE.  Use  the  foot-pound-second 
system  of  units.  Assume  a  convenient  mean  diameter  of  fly- 
wheel rim.  From  this  and  the  given  maximum  rotative  speed 
of  the  fly-wheel  shaft  find  vif  Solve  the  above  equation  for 
Wthus: 

2gAE 


Substitute  the  values  of  AE,  vi,  v2,  and  £=32.2  feet  per  second2, 
whence  W  becomes  known,  =  weight  of  fly-wheel  rim.  The 
weight  of  rim  only  will  be  considered;  the  other  parts  of  the 
wheel,  being  nearer  the  axis,  have  less  velocity  and  less  capacity 
per  pound  for  storing  energy.  Their  effect  is  to  reduce  slightly 
the  allowable  variation  of  velocity.* 

189.  Problem. — In    a   punching-machine  the  belt  is  capable 
of  applying  a  uniform  torsional  effort  to  the  shaft;    but  most 

*  Numerical  examples  taken  from  ordinary  medium-sized  steam-engine  fly- 
wheels show  that  while  the  combined  weight  of  arms  and  hub  equals  about  one 
third  of  the  total  weight  of  the  wheel,  the  energy  stored  in  them  for  a  given  varia- 
tion of  velocity  is  only  about  10  per  cent  of  that  stored  in  the  rim  for  the  same 
variation. 


330  MACHINE  DESIGN. 

of  the  time  it  is  only  required  to  drive  the  moving  parts  of  the 
machine  against  frictional  resistance.  At  intervals,  however, 
the  punch  must  be  forced  through  metal  which  offers  shearing 
resistance  to  its  action.  Either  the  belt  or  fly-wheel,  or  the  two 
combined,  must  be  capable  of  overcoming  this  resistance.  A 
punch  makes  30  strokes  per  minute,  and  enters  the  die  \  inch.  It 
is  required  to  punch  J-inch  holes  in  steel  plates  J  inch  thick. 
The  shearing  strength  of  the  steel  is  about  50,000  Ibs.  per  square 
inch.  When  the  punch  just  touches  the  plate  the  surface  which 
offers  shearing  resistance  to  its  action  equals  the  surface  of  the 
hole  which  results  from  the  punching,  =ndt,  in  which  d  =  diam- 
eter of  hole  or  punch,  /  =  thickness  of  plate.  The  maximum 
shearing  resistance,  therefore,  equals  7^X^X50,000  =  58,900  Ibs. 
As  the  punch  advances  through  the  plate  the  resistance  decreases 
because  the  surface  in  shear  decreases,  and  when  the  punch 
just  passes  through  the  plate  the  resistance  becomes  zero.  If 
the  change  of  resistance  be  assumed  uniform  (which  would 
probably  be  approximately  true)  the  mean  resistance  to  punching 
would  equal  the  maximum  resistance  +  minimum  resistance,  -5- 2, 

=  — =29,450.     The  radius  of  the  crank  which  actuates 

the  punch  =  2  in.  In  Fig.  187  the  circle  represents  the  path  of 
the  crank-pin  center.  Its  vertical  diameter  then  represents  the 
travel  of  the  punch.  If  the  actuating  mechanism  be  a  slotted 
cross-head,  as  is  usual,  it  is  a  case  of  harmonic  motion,  and  it  may 
be  assumed  that  while  the  punch  travels  vertically  from  A  to  B, 
the  crank-pin  center  travels  in  the  semicircle  ACB.  Let  BD 
and  DE  each=J  inch.  Then  when  the  punch  reaches  E  it 
just  touches  the  plate  to  be  punched,  which  is  J  inch  thick,  and 
when  it  reaches  D  it  has  just  passed  through  the  plate.  Draw  the 
horizontal  lines  EF  and  DG  and  the  radial  lines  OG  and  OF. 
Then,  while  the  punch  passes  through  the  plate,  the  crank-pin 
center  moves  from  F  to  G,  or  through  an  angle  (in  this  case) 


FLY-WHEELS  AND   PULLEYS. 


331 


of  19°.  Therefore  the  crank-shaft  A,  Fig.  188,  and  attached 
gear  rotate  through  19°  during  the  action  of  the  punch.  The 
ratio  of  angular  velocity  of  the  pinion  and  the  gear  =  the  inverse 

ratio  of  pitch  diameters  = —  =  5.     Hence   the   shaft  B   rotates 

through  an  angle  =i9°X5  =95°  during  the  action  of  the  punch. 
If  there  were  no  fly-wheel  the  belt  would  need  to  be  designed 
to  overcome  the  maximum  resistance;  i.e.,  the  resistance  at  the 
instant  when  the  punch  is  just  beginning  to  act.  This  would 


rn 

I 

I 
£ 

! 

L 

1  Loose  Pulleys  j 

i 

Flywheel 

II 

Crank                      A 
B 

nr! 

i 

FIG.  188. 


give  for  this  case  a  double  belt  about  20  inches  wide.  The  need 
for  a  fly-wheel  is  therefore  apparent.  Assume  that  the  fly- 
wheel may  be  conveniently  36  inches  mean  diameter,  and  that 
a  single  belt  5  inches  wide  is  to  be  used.  The  allowable  maxi- 
mum tension  is  then  =5Xallowable  tension  per  inch  of  width 
of  single  belting  =  5  X  70  =  350  Ibs.  =  TV 

Since  the  pulley-shaft  makes  150  revolutions  per  minute  and 
the  diameter  of  the  pulley  is  2  feet,  the  velocity  of  the  belt  = 
150X2X^  =  942  feet  per  minute.  At  this  slow  speed  the  simple 

rri 

form  of  the  belt  formula  may  be  used,  i.e.,  log  ^r=  0.4343 /£0. 

•*  2 

Assume  an  angle  of  contact  of  180°.     Then 


33 2  MACHINE  DESIGN. 

#  =  3.1416, 


T-" 

log  7^=0.4081; 

^  2 

and  /.  — -  =  2.56. 


T\  —  T%  =  213.3  Ibs.  =  the  driving  force  at  the  surface  of  the  pulley. 
Assume  that  the  frictional  resistance  of  the  machine  is 
equivalent  to  25  Ibs.  applied  at  the  pulley-rim.  Then  the  belt 
can  exert  213.3  —  25=188.3  Ibs.  =P,  to  accelerate  the  fly-wheel 
or  to  do  the  work  of  punching.  Assume  variation  of  velocity  =  10 
per  cent.  The  work  of  punching  =  the  mean  resistance  offerpd 
to  the  punch  multiplied  by  the  space  through  which  the  punch 

acts,   =  —   —X  0.5"  =  1472  5  in.  -Ibs.  =1225  ft.  -Ibs.     The  pulley- 

shaft  moves  during  the  punching  through  95°,  and  the  driv- 
ing tension  of  the  belt,  =P  =  188.3  Ibs.,  does  work  =  PXspace 

moved  through  during  the  punching  =  188.3  Ibs.  X7r^—r-  =  188.3 

lbs.X?rX2    ft.  X-^r-  =  312    ft.-lbs.     The   work  left   for   the    fly- 
300 

wheel  to  give  out  with  a  reduction  of  velocity  of  10  per  cent 
=  1225—312=913  ft.-lbs.  Let  vi=  maximum  velocity  of  fly- 
wheel rim;  7;2=  minimum  velocity  of  fly-wheel  rim;  W  =  weight 
of  the  fly-wheel  rim.  The  energy  it  is  capable  of  giving  out, 


while  its  velocity  is  reduced  from  v\  to  v2,   =  -  -  —  ,  anJ 

the  value  of  W  must  be  such  that  this  energy  given  out  shall 
equal  913  ft.-lbs.     Hence  the  following  equation  may  be  written  : 


FLY-WHEELS  AND  PULLEYS.  333 


Therefore 


The  punch-shaft  makes  30  revolutions  per  minute  and  the  pulley- 
shaft    30X5  =  150  =-ZV   revolutions   per  minute.      Hence    Vi    in 

feet  per  second=  —7  —  ,  D  being  fly-wheel  diameter  in  feet  =3  feet. 
150X37: 

vi-  -^-  =23.56; 

=21.2; 


Hence  W  =  y-  ^^   -  =555  Ibs. 

To  proportion  the  rim:    A  cubic  inch  of  cast  iron  weighs 
0.26  lb.;   hence  there  must  be  ^^  =  2135  cu.  ins.     The  cubic 

contents  of  the  rim=mean  diameter  XrcX  its  cross- sectional  area 
A  =2135  cu.  ins.;  hence 

>T  -5  f 

=  18.45  sq.  ii 


If  the  cross-section  were  made  square  its  side  would  =\/ 18.45 

=  4.3- 

190.  Pump  Fly-wheel. — The  belt  for  the  pump,  p.  304,  is 
designed  for  the  average  work.  A  fly-wheel  is  necessary  to 
adapt  the  varying  resistance  to  the  capacity  of  the  belt.  The 
rate  of  doing  work  on  the  return  stroke  (supposing  no  resistance 
due  to  suction)  is  only  equal  to  the  frictional  resistance  of  the 
machine.  During  the  working  stroke  the  rate  of  doing  work 
varies  because  the  velocity  of  the  plunger  varies,  although  the 
pressure  is  constant.  The  rate  of  doing  work  is  a  maximum 
when  the  velocity  of  the  plunger  is  greatest.  In  Fig.  189,  A 
is  the  velocity  diagram,  B  is  the  force  diagram,  C  is  the 


334 


MACHINE  DESIGN. 


tangential  diagram  drawn  as  indicated  on  pp.  78-80.  The 
belt,  5  inches  wide,  is  capable  of  applying  a  tangential  force 
of  148  Ibs.  to  the  1 8-inch  pulley- rim.  The  velocity  of  the  pulley- 
rim  =  711.5X300  =  1414'.  The  velocity  of  the  crank-pin  axis 
=  7rX  0.833X50  =  130. 9'.  Therefore  the  force  of  148  Ibs.  at 

the  pulley- rim  corresponds  to  a  force   =i48x — — -  =  I=;QO   Ibs. 

130.9 

applied  tangentially  at  the  crank-pin  axis.  This  may  be  plotted 
as  an  ordinate  upon  the  tangential  diagram  C,  from  the  base 


FIG.  189. 


line  XXi,  using  the  same  force  scale.  Through  the  upper 
extremity  of  this  ordinate  draw  the  horizontal  line  DE.  The 
area  between  DE  and  XXi  represents  the  work  the  belt  is 
capable  of  doing  during  the  working  stroke.  During  the  return 
stroke  it  is  capable  of  doing  the  same  amount  of  work.  But 
this  work  must  now  be  absorbed  in  accelerating  the  fly-wheel. 
Suppose  the  plunger  to  be  moving  in  the  direction  shown  by 
the  arrow.  From  E  to  F  the  effort  is  in  excess  and  the  fly-wheel 
is  storing  energy.  From  F  to  G  the  resistance  is  in  excess  and 
the  fly-wheel  is  giving  out  energy.  The  work  the  fly-wheel  must 
be  capable  of  giving  out  with  the  allowable  reduction  of  velocity 
is  that  represented  by  the  area  under  the  curve  above  the  line 
FG.  From  G  to  D,  and  during  the  entire  return  stroke,  the 
belt  is  doing  work  to  accelerate  the  fly-wheel.  This  work 


FLY-WHEELS  /IND   PULLEYS.  335 

becomes  stored  kinetic  energy  in  the  fly-wheel.  Obviously  the 
following  equation  of  areas  may  be  written : 

Xi  EF+  XGD  +  XHKX  i  =  GMF. 

The  left-hand  member  of  this  equation  represents  the  work 
done  by  the  belt  in  accelerating  the  fly-wheel;  the  right-hand 
member  represents  the  work  given  out  by  the  fly-wheel  to  help 
the  belt. 

The  work  in  foot-pounds  represented  by  the  area  GMF 
may  be  equated  with  the  difference  of  kinetic  energy  of  the 
fly-wheel  at  maximum  and  minimum  velocities.  To  find  the 
value  of  this  work:  One  inch  of  ordinate  on  the  force  diagram 
represents  8520  Ibs.;  one  inch  of  abscissa  represents  0.449  foot. 
Therefore  one  square  inch  of  area  represents  8520  Ibs.  Xo.449' 
=  3825.48  ft.-lbs.  The  area  GMF  =0.4  sq.  ins.  Therefore 
the  work  =382 5. 48X0.4  =  1530  ft.-lbs.  =  JE.  The  difference  of 

W 

kinetic  energy  =—  (vi2—v22)  =  1530;     W   equals  the  weight   of 

the  fly-wheel  rim.     Hence 

w_  1 530X32.2X2 

Assume  the  mean  fly-wheel  diameter  =  2. 5  feet.  It  will  be 
keyed  to  the  pulley-shaft,  and  will  run  300  revolutions  per 
minute,  =  5  revolutions  per  second.  The  maximum  velocity 
of  fly-wheel  rim  =71X2.5X5  =39.27  =v\.  Assume  an  allowable 
variation  of  velocity,  =5  per  cent.  Then  ^2=37-27Xo.95  = 
37.3;  ^12  =  1542.3;  1/2  =  1391.3;  Vi2-v22=i$i.  Hence 

f  TF  =  I53OX32.2X2=65ilbs> 

There  must  be  651  -^0.26  cu.  in.  in  the  rim,  =2504.  The  mean 
circumference  =  3o"X?r  =  94. 2".  Hence  the  cross-sectional  area 


336  MACHINE  DESIGN. 

of    rim  =2504-^94.2  =26.6  sq.    ins.      The   rim   may   be    made 
4-5"  X6". 

The  frictional  resistance  of  the  machine  is  neglected.  It 
might  have  been  estimated  and  introduced  into  the  problem  as  a 
constant  resistance. 

191.  Steam-engine  Fly-wheel. — From   given   data   draw  the 
indicator-card  as  modified  by  the  acceleration  of  reciprocating 
parts.     See  page  77  and  Fig.  46.      From  this  and  the  velocity 
diagram   construct    the   diagram   of   tangential    driving   force, 
Fig.  47.    Measure  the  area  of  this  diagram  and  draw  the  equiva- 
lent rectangle  on  the  same  base.     This  rectangle  represents  the 
energy  of  the  uniform  resistance  during  one  stroke;    while  the 
tangential  diagram  represents  the  work  done  by  the  steam  upon 
the    crank-pin.     The    area    of    the    tangential   diagram    which 
extends  above  the  rectangle  represents  the  work  to  be  absorbed 
by  the  fly-wheel  with  the  allowable  variation  of  velocity.*     Find 
the  value  of  this  in  foot-pounds,  and  equate  it  to  the  expression 
for  difference   of    kinetic   energy  at  maximum   and  minimum 
velocity.     Solve  for  W,  the  weight  of  fly-wheel. 

192.  Stresses  in  Fly-wheel   Rims. — Mathematical   analyses 
of  the  stresses  in  fly-wheel  rims  are  unsatisfactory.     In  the  first 
place,  in  order  to  get  solutions  of  reasonable   simplicity  it   is 
customary   to   make    assumptions   which    are   contrary   to   the 
actual   conditions;    and    in   the   second   place,   no   satisfactory 
data  exist  concerning  the  strength  of  cast  iron  in  such  heavy 
sections  as  are  used  in  large  engine  fly-wheels.     An  examination 
of  the  nature  of  the  stresses,  however,  will  indicate  the  points 
to  be  looked  out  for  in  design. 

Considering  a  ring  of  hollow  cylindrical  form,  comparatively 

*  For  compound  engines  and  for  varying  resistances  the  dia  grams  should  be 
constructed  for  the  complete  cycle.  For  full  treatment  of  the  problem  of  fly 
wheels  for  engines  driving  alternators  the  reader  is  referred  to  the  Trans.  A.  S. 
M.  E.,  Vol.  XXII,  p.  955,  and  Vol.  XXIV,  p.  98. 


FLY-WHEELS  AND  PULLEYS.  337 

thin  radially,  it  can  be  shown  that,  when  it  is  rotated  about  its 
axis,  tension  is  set  up  in  the  ring  proportional  to  the  weight  of 
the  material  used  and  the  square  of  the  linear  velocity.  This 
tension  is  due  solely  to  the  action  of  "  centrifugal  force"  and  is 
termed  "centrifugal  tension." 

Consider  the  half-  ring  shown  in  Fig.  190: 
v=  the  velocity  of  the  rim  in  feet  per  second. 
c  =  "centrifugal  force"  per  foot  of  rim; 
R=  radius  in  feet; 
A  =area  of  rim  in  square  inches; 
P=  total  tension  in  rim  in  pounds; 
ft  =unit  tensile  stress  in  rim  in  pounds; 
iv  =  weight   of  material   as  represented  by  a  piece  i  inch 

square  and  i  foot  long; 
g  =32.2  feet  per  second  per  second; 
2P=sum  of  horizontal  components  of  all  the  small  centri- 

fugal forces  cds\ 

Each  horizontal  component  =cds  cos  6,  which  may  be  written 
cR  cos  d  dd,  because  ds  =  Rd6. 


2P= 


But,  c  = 


.'.    P  =  cR. 

Mv2     W  v2 


R       gR9 

•'•  p=  v> 

W  being  the  weight  of  one  linear  foot  of  rim=m4 
Also,  P=ftA-, 

_W  2_wAv2 

~  gV         g     ' 

iw2 


338  MACHINE  DESIGN. 

For  cast  iron,  putting  ft  =  20,000  Ibs.,  the  ultimate  strength, 
and  7^  =  0.26X12  Ibs.,  it  follows  that  -^  =  454  feet  per  second. 
In  other  words  a  cast-iron  ring  will  burst  at  a  speed  of  454  feet 
per  second.  Furthermore,  an  examination  of  the  formula 
shows  that  for  a  ring  this  bursting  velocity  depends  not  at  all 
on  the  size  or  shape  of  the  cross-section,  but  only  on  the  material 
used  as  represented  by  ft  and  w.  This  is  not  entirely  true.  In 
La  cylindrical  disk,  without  any  hole,  the  maximum  centrifugal 

tension  is  at  the  center,  where  ft  =  —i       — .     For  a  cylindrical 

o 

disk  with  incipient  hole,  ft=—       — .     For  the  derivation  of  these 

g 
formulae,  see  Ewing's  Strength  of  Materials;  also  Stodola's  Steam 

Turbines* 

j;  This  centrifugal  tension  causes  a  corresponding  elongation  of 
the  material  and  therefore  an  increase  in  the  radius  of  the  ring.  A 
free,  thin  ring  of  whatever  cross-section  can  and  does  take  the  new 
radius  and  the  tension  on  all  sections  =  /,  pounds  per  square  inch. 

With  the  introduction  of  rigidly  fastened  arms  a  number  of 
new  and  vital  elements  enter  into  the  problem.  An  arm  of  the 
same  original  length  as  the  original  radius  of  the  rim  when 
rotated  about  an  axis  perpendicular  to  its  inner  end  will  also 
suffer  an  elongation  due  to  centrifugal  action.  The  amount  of  this 
radial  elongation  will  vary  with  the  form  of  the  arm,  but  in  no 
practical  case  will  it  amount  to  as  much  as  one  third  of  the 
radial  increase  of  the  ring  rotating  at  the  same  speed. 

To  accommodate  this  difference  the  arm,  if  rigidly  fastened 
to  hub  and  rim,  will  be  extended  lengthwise  by  the  rim  and  the 
rim  will  be  drawn  in,  out  of  its  regular  circular  form,  by  the 
arm.  The  relation  between  the  amount  the  arm  is  drawn  out 
and  the  amount  the  rim  is  drawn  in  is  governed  by  the  propor- 
tions of  these  parts. 

*  See  also,  Moss  in  Trans.  A.  S.  M.  E.,  Vol.  XXXIV. 


FLY-WHEELS  AND   PULLEYS.  339 

The  result  is  that  the  rim  tends  to  bow  out  between  the  arms 
and  really  become  akin  to  a  uniformly  loaded  continuous  beam 
with  the  dangerous  sections  midway  between  the  arms  and  at 
the  points  of  junction  of  arms  and  rim.  The  fallacy  of  applying 
the  ring  theory  solely  to  the  fly-wheel  rim  becomes  evident  at 
once.  In  a  free  ring  the  form  of  cross-section  is  immaterial,  as 
the  section  is  subjected  only  to  tension.  In  the  rim  with  arms 
the  form  of  cross-section  becomes  a  vital  point,  as  the  rim  is 
subjected  to  flexure  as  well  as  tension,  and  the  strength  of  a 
member  to  resist  flexure  depends  directly  upon  the  modulus 
of  the  section. 

In  addition  to  the  foregoing  stresses,  which  are  induced 
under  all  conditions,  even  under  the  extreme  supposition  that 
the  wheel  is  rotating  at  a  perfectly  uniform  rate,  there  are  others 
when  the  rim  is  considered  as  performing  its  functions — i.e., 
in  a  balance-wheel,  absorbing  or  giving  out  energy  by  changes 
of  velocity  and,  in  a  band-wheel,  transmitting  the  power. 

This  may  be  seen  by  reference  to  Fig.  191.  A  shows  the 
relation  between  rim,  arm,  and  hub  when  the  wheel  is  at  rest 
or  rotating  uniformly  and  not  transmitting 
any  power.  B  shows  the  relation  when 
work  is  being  done.  The  arm  becomes  an 
encastre  beam  and  corresponding  stresses 
are  induced  in  it.  Furthermore,  the  bend- 
ing of  the  arm  tends  to  shorten  it  radially,  thus  drawing  in  the 
outer  end,  which  increases  the  flexure  in  the  rim.  In  addition 
to  the  foregoing  there  are  stresses  in  the  rim  due  to  the  weight 
of  the  wheel,  shrinkage,  etc.,  which  cannot  be  eliminated. 

193.  Stresses  in  Arms  of  Puljeys  or  Fly-wheels. — The  arms 
are  principally  stressed  by  the  bending  moment  due  to  variations 
of  velocity  of  the  wheel  or  to  the  power  transmitted. 

Let  Mt  =  the  greatest  turning  moment  transmitted  in  inch- 
pounds; 


340  MACHINE  DESIGN. 

n=  number  of  arms; 

}t  =safe  unit  stress  in  outer  fiber  of  arm  in  pounds  per 
square  inch; 

—  =  modulus  of  section  of  arm,  dimensions  in  inches. 
c 

Then 

Mt  =  nft—  may  be  written  and  solved  for—. 

Having  determined  upon  the  form  of  cross-section  the  dimensions 

can  be  determined  from  this  value  of—. 

c 

If  Mt  is  unknown  the  arms  can  be  made  as  strong  as  the  shaft 
by  equating  the  twisting  strength  of  the  shaft  to  the  bending 
strength  of  the  arms,  thus: 


f  «•* 

Is ==  illt —  • 

2  C 

;i  =  allowable  shearing  stress  in  outer  fiber  of  shaft,  pounds  per 

square  inch; 
r  =  radius  of  shaft  in  inches; 

n,  }tj  and  —  as  before. 
c 

Consider  junction  of  arm  and  hub  next.     (See 
Fig.  192.) 

The  tendency  for  the  arm  to  fail  through 
flexure  on  the  section  A-A  may  be  equated  to  the  tendency 
for  the  bolts  2  and  3  to  shear  off,  using  i  as  a  pivot. 

Let  A  =  combined  shearing  areas  of  2  and  3,  square  inches; 
j8=  allowable  shearing  stress  of  2  and  3  in  pounds  per 

square  inch; 

/^distance  between  centers  i  and  2,  and    i  and  3,  in 
inches; 


FLY- WHEELS  AND  PULLEYS.  341 

jt=  allowable  stress  in  outer  fiber  of  arm   in  pounds  per 
square  inch; 

—  =  modulus  of  arm  section,  dimensions  in  inches. 
c 


Then 


which  can  be  solved  for  A,  the  desired  area. 

If  the  arm  is  bolted  to  the  rim  a  similar  method  may  be  em- 
ployed to  make  the  bolts  as  strong  as  the  arm. 

194.  Construction  of  Fly-wheels. — Since  weight  is  so  great 
a  factor  in  fly-wheels  it  has  been  common  practice  to  make 
them  of  that  material  which  combines  greatest  weight  with 
least  cost,  namely,  cast  iron.  That  this  is  not  always  safe 
practice  has  been  conclusively  demonstrated  by  many  serious 
accidents. 

Up  to  10  feet  in  diameter  the  wheels  are  generally  cast 
in  a  single  piece.  Occasionally  the  hub  is  divided  to  relieve 
the  stress  due  to  cooling.  In  such  cases,  supposing  the  wheel 
to  have  six  arms,  the  hub  is  made  in  three  sections,  each  having 
a  pair  of  arms  running  to  the  rim.  Since  the  sections  are  inde- 
pendent, any  pair  of  arms  can  adjust  itself  to  the  conditions  of 
shrinkage  without  subjecting  the  other  arms  to  indeterminate 
stresses.  The  hub  sections  are  separated  from  each  other  by 
a  space  of  half  an  inch  or  less  and  this  is  filled  with  lead 
or  babbitt  metal.  Then  shrink-rings  or  bolts  are  used  to  hold 
the  sections  together.  Sometimes  the  hub  is  only  split  into  two 
parts. 

For  reasons  connected  chiefly  with  transportation,  wheels 
from  10  to  15  feet  in  diameter  are  cast  in  two  halves  which 
are  afterwards  joined  together  by  flanges  and  bolts  at  the  rim, 
and  shrink-rings  or  bolts  at  the  hub. 


342  MACHINE  DESIGN. 

In  still  larger  and  heavier  wheels  the  hub  is  generally  made 
entirely  separate  from  the  arms.  The  rim  is  made  in  as  many 
segments  as  there  are  arms.  Sometimes  the  arm  is  cast  with 
the  segment  and  sometimes  the  arms  and  segments  are  cast 
separately.  The  hub  is  commonly  made  in  the  form  of  a  pair 
of  disks  having  a  space  between  them  to  receive  the  arms  which 
are  fastened  to  them  by  means  of  accurately  fitted  through  bolts. 

Unless  the  wheel  is  to  be  a  forced  fit  on  its  shaft  it  is  best 
to  have  three  equally  spaced  keyways,  so  that  it  may  be  kept 
accurately  centered  with  the  shaft. 

In  these  large  wheels  the  joints  of  the  segments  of  the  rim 
are  usually  midway  between  the  arms  and  steel  straps  or  links 

suc^  as  are  snowri  in  Fig.  193  are 
heated  and  dropped  into  recesses  pre- 
FlGt  I93<  viously  fitted  to  receive  them.  As  they 

cool,  their  contraction  draws  the  joint  together.  They  should 
not,  however,  be  subjected  to  a  very  great  initial  tension  of  this 
sort.  The  form  shown  at  A  is  most  commonly  used.  The  links 
are  made  of  high-grade  steel  and  their  area  is  such  that  their  ten- 
sile strength  equals  that  of  the  reduced  section  of  the  rim.  The 
areas  subjected  to  shear  and  compression  must  also  have  this 
strength. 

Taking  the  nature  of  the  stresses  into  consideration  it  is  clear 
that  the  rim  should  always  be  as  deep  radially  as  possible  to 
resist  the  flexure  action,  also  that  the  arms  should  be  near  together. 
Many  arms  are  much  better  than  a  few  and  a  disk  or  web  is  still 
better.* 

The  strongest  wheel  having  arms  will  be  one  whose  rim  is 
cast  in  a  single  piece,  while  the  arms  and  hubs  are  cast  as  a  second 
piece.  On  the  inside  of  the  rim  there  are  lugs  between  which 

*  Disk  wheels  have  the  further  advantage  of  offering  less  resistance  to  the 
air.  This  maybe  a  considerable  item.  See  Cassier's  Mag.,  Vol.  23,  pp.  577  and 
76i. 


FLY-WHEELS   AND  PULLEYS,  343 

the  ends  of  the  arms  fit  so  that  there  is  a  space  of  about  one 
fourth  inch  all  around.  (See  Fig.  194.)  This  space  may  be 
filled  with  oakum  well  driven  in.  It  is  clear  that  the  rim  in  this 
case  acts  as  a  free  ring  and  is  subjected  solely  to  centrifugal 
tension  .* 

Joints  in  the  rim  must  always  be  a  source  of  weakness  whether 
located  at  the  end  of  the  arms  or  midway  between  arms. 

V-q^prV 

FIG.  194.  FIG.  195. 

If  a  flanged  joint  midway  between  the  arms  is  used,  such  as 
is  shown  in  Fig.  195,  which  is  common  practice  for  split  band- 
wheels  of  medium  size,  the  flanges  should  be  deep  radially  and 
well  braced  by  ribs.  The  bolts  should  be  as  close  to  the  rim 
as  possible,  and  a  tension  rod  should  carry  the  extra  stress  (due 
to  the  weight  of  the  heavy  joint  and  its  velocity)  to  the  hub. 
Experiments  made  by  Prof.  C.  H.  Benjamin  f  show  that  the  use 
of  such  tie-rods  increases  the  strength  of  the  wheel  100  per  cent 
over  that  of  a  similar  wheel  without  tie-rods.  He  also  found 
that  jointed  rims  are  only  one  fourth  as  strong  as  solid  rims. 

Probably  as  strong  a  form  of  cast-iron  built-up  wheel  for 
heavy  duty  as  any  yet  designed  is  one  described  by  Mr.  John 
Fritz,!  having  a  hollow  rim  and  many  arms. 

But,  at  the  best,  cast  iron  is  an  uncertain  material  to  use 
for  such  tensile  and  flexure  stresses  as  are  induced  in  a  heavy- 
duty  fly-wheel,  and  it  is  wiser  to  make  such  wheels  of  structural 
steel.  A  built-up  wheel  having  a  disk  or  web  of  steel  plates  and 
a  rim  of  the  same  material,  all  joints  being  carefully  "  broken  " 
and  strongly  riveted,  is  so  much  better  than  any  built-up  cast- 

*  See  Trans.  A.  S.  M.  E.,  Vol.  XX,  p.  944,  and  Vol.  XXI,  p.  322. 
t  Ibid.,  Vols.  XX  and  XXIII. 
$  Ibid.,  Vol.  XXI. 


344  MACHINE  DESIGN. 

iron  wheel  that  the  latter  are  passing  out  of  use.  The  steel 
wheels  can  have  at  least  twice  the  rim  velocity  of  the  cast  wheels 
with  greater  safety  and  may  therefore  be  much  lighter  for  the 
same  duty.  Their  lesser  weight  makes  less  pressure  on  the 
bearings  and  consequently  less  friction  loss.* 

Plate  II  shows  forms  of  rim  joints  for  split  rim  flywheels 
and  pulleys  which  are  probably  as  strong  as  any  that  can  be 
devised.  They  are  taken  from  the  American  Machinist ,  Vol.  30. 
It  is  a  mistake,  however,  to  believe  that  any  of  these  joints  will 
will  give  as  strong  a  wheel  as  one  having  a  solid  rim. 

Pulley  rims,  according  to  Carl  Earth,  should  have  a  mini- 
mum width,  F,  equal  to  i^-  times  the  belt  width  +  ^  inch.  Their 
thickness  is  a  question  of  sound  castings  and  avoidance  of  shrink- 
age stresses  rather  than  of  strength  which  can  be  computed. 
The  minimum  finished  thickness  at  the  edge  of  pulleys  6  or  8 
inches  in  diameter  may  be  as  little  as  |  inch,  rising  to  f  inch 
for  72  inches  diameter.  The  radial  height  of  crown  may  be 

F% 

— .     The  diameter  of  hub  may  be  made  twice  the  bore.     Six 

32 

arms  of  elliptical  cross-section  are  most  frequently  employed, 
the  minor  axis  being  about  one-half  the  major.  To  get  the  width 
of  the  arm  at  the  hub,  the  circumference  of  the  latter  is  divided 
into  as  many  equal  parts  as  there  are  arms.  The  arms  are 
tapered  in  width,  about  J  inch  per  foot,  but  may  have  uniform 
thickness.  Generous  fillets  should  be  employed  where  they 
join  each  other  and  the  hub,  as  well  as  where  they  join  the  rim. 
Transition  from  thick  to  thin  sections  must  be  made  as  gradual 
as  possible.  For  extra  width  of  face  two  parallel  sets  of  arms 
may  be  used. 

*  For  drawings  and  descriptions  of  wheels  made  of  forged  materials  the  reader 
is  referred  to  Vol.  XVII,  Trans.  A.  S.  M.  E.,  and  Power,  April  1894,  Nov.  1895, 
Jan.  1896,  and  Nov.  1897.  Also,  Jones'  Machine  Design. 


PLATE  II. 


1 

p 

I 

j 

0         J 

i"  °            I 

~. 

0        j 

° 

fl 

HAIGHT'S  JOINT  FOR  HEAVY  RIM. 


FLANGED  JOINT   OVER-ARM 
SEGMENTAL  RIM. 


o 


DOUBLE  ARM   JOINT  .FOR    WIDE  RIM. 


DOUBLE  ARM  JOINT  FOR  SHEAVE  WHEEL. 


345 


CHAPTER  XVII. 

TOOTHED  WHEELS   OR   GEARS. 

195.  Fundamental  Theory  of  Gear  Transmission.  —  When 
toothed  wheels  are  used  to  communicate  motion,  the  motion 
elements  are  the  tooth  surfaces.  The  contact  of  these  surfaces 
with  each  other  is  line  contact.  Such  pairs  of  motion  elements 
are  called  higher  pairs,  to  distinguish  them  from  lower  pairs, 
which  are  in  contact  throughout  their  entire  surface.  Fig.  196 
shows  the  simplest  toothed-wheel  mechanism.  There  are  three 
links,  a,  b,  and  c,  and  therefore  three  centros,  ab,  be,  and  ac.  These 
centros  must,  as  heretofore  explained,  lie  in  the  same  straight 
line,  ac  and  ab  are  the  centers  of  the  turning  pairs  connecting 
c  and  b  to  a.  It  is  required  to  locate  be  on  the  line  of  centers. 

When  the  gear  c  is  caused  to  rotate  uniformly  with  a  certain 
angular  velocity,  i.e.,  at  the  rate  of  m  revolutions  per  minute, 
it  is  required  to  cause  the  gear  b  to  rotate  uniformly  at  a  rate 
of  n  revolutions  per  minute.  The  angular  velocity  ratio  is  there- 

fore constant   and  =  —  .     The  centre  be  is  a  point  on  the  line  of 
n 

centers  which  has  the  same  linear  velocity  whether  it  is  con- 
sidered as  a  point  in  b  or  c.  The  linear  velocity  of  this  point  be 
in  b  =  2nRin',  and  the  linear  velocity  of  the  same  point  in  c  =  2xR2m  ; 
in  which  RI  =  radius  of  be  in  b,  and  ^2=  radius  of  be  in  c.  But 
this  linear  velocity  must  be  the  same  in  both  cases,  and  hence  the 
above  expressions  may  be  equated  thus: 


347 


348 
whence 


MACHINE  DESIGN. 


Hence  be  is  located  by  dividing  the  line  of  centers  into  parts  which 

are  to  each  other  inversely  as  the  angular  velocities  of  the  gears. 

Thus,  let  ab  and  ac,  Fig.  197,  be  the  centers  of  a  pair  of  gears 


m 


whose  angular  velocity  ratio  =—.      Draw  the   line   of    centers; 

divide  it  into  m  +  n  equal  parts;  m  of  these  from  ab  toward  the 
right,  or  n  from  ac  toward  the  left,  will  locate  be.  Draw  circles 
through  be,  with  ab  and  ac  as  centers.  These  circles  are  the 
centrodes  of  be  and  are  called  pitch  circles.  It  has  been  already 
explained  that  any  motion  may  be  reproduced  by  rolling  the 
centrodes  of  that  motion  upon  each  other  without  slipping. 


FIG.  196. 


•  FIG.  197. 


Therefore  the  motion  of  gears  is  the  same  as  that  which  would 
result  from  the  rolling  together  of  the  pitch  circles  (or  cylinders) 
without  slipping.  In  fact,  these  pitch  cylinders  themselves 
might  be,  and  sometimes  are,  used  for  transmitting  motion  of 
rotation.  Slipping,  however,  is  apt  to  occur,  and  hence  these 
"friction-gears"  cannot  be  used  if  no  variation  from  the  given 
velocity  ratio  is  allowable.  Hence  teeth  are  formed  on  the 
wheels  which  engage  with  each  other,  to  prevent  slipping. 

196.  Definitions. — If  the  pitch  circle  be  divided  into  as  many 
equal  parts  as  there  are  teeth  in  the  gear,  the  arc  included  between 


TOOTHED   WHEELS   OR    GEARS.  349 

two  of  these  divisions  is  the  circular  pitch  *  of  the  gear.  Circular 
pitch  may  also  be  defined  as  the  distance  on  the  pitch  circle 
occupied  by  a  tooth  and  a  space ;  or,  otherwise,  it  is  the  distance 
on  the  pitch  circle  from  any  point  of  a  tooth  to  the  corresponding 
point  in  the  next  tooth.  A  fractional  tooth  is  impossible,  and 
therefore  the  circular  pitch  must  be  such  a  value  that  the  pitch 
circumference  is  divisible  by  it.  Let  P=  circular  pitch  in  inches; 
let  D=  pitch  diameter  in  inches;  N=  number  of  teeth;  then 

NP=nD;    AT— 5-;    D=     -;    p==Jj-     From   these   relations 

any  one  of  the  three  values,  P,  D,  and  N,  may  be  found  if  the 
other  two  are  given. 

Diametral  pitch  is   the  number  of    teeth  per  inch  of  pitch 

N 
diameter.     Thus  if  />=diametral  pitch,  p=j^-     Multiplying  the 

nD  N  xD    N 

two  expressions,  P=~*r  anc*  P  =~n>  together  gives  Pp  =^v7  •  77  =7r- 

Or,  the  product  of  diametral  and  circular  pitch  =n.  Circular 
pitch  is  usually  used  for  large  cast  gears,  and  for  mortise-gears 
(gears  with  wooden  teeth  inserted) .  Diametral  pitch  is  usually 
used  for  small  cut  gears. 

In  Fig.  198,  b,  c,  and  k  are  pitch  points  of  the  teeth;  the  arc 
bk  is  the  circular  pitch;  ab  is  the  face  of  the  tooth;  bm  is  the  flank 
of  the  tooth;  the  whole  curve  abm  is  the  profile  of 
the  tooth;  AD  is  the  total  depth  of  the  tooth;  AC 
is  the  working  depth]  AB  is  the  addendum ;  a 
circle  through  A  is  the  addendum  circle.  Clear- 
ance is  the  excess  of  total  depth  over  working 
depth,  =CD.  Backlash  is  the  width  of  space  on  the  pitch  line 
minus  the  width  of  the  tooth  on  the  same  line.  In  cast  gears 
whose  tooth  surfaces  are  not  "tooled,"  backlash  needs  to  be 

*  Sometimes  called  circumferential  pitch. 


35°  MACHINE  DESIGN. 

allowed,  because  of  unavoidable  imperfections  in  the  surfaceSc 
In  cut  gears,  however,  it  may  be  reduced  almost  to  zero,  and  the 
tooth  and  space,  measured  on  the  pitch  circle,  may  be  considered 
equal. 

197.  Conditions  Governing  Forms  of  Teeth. — Teeth  of  almost 
any  form  may  be  used,  and  the  average  velocity  will  be  right. 
But  if  the  forms  are  not  correct  there  will  be  continual  variations 
of  velocity  ratio  between  a  minimum  and  maximum  value.     These 
variations  are  in  many  cases  unallowable,  and  in  all  cases  unde- 
sirable.    It  is  necessary  therefore  to  study  tooth  outlines  which 
shall  serve  for  the  transmission  of  a  constant  velocity  ratio. 

The  centre  of  relative  motion  of  the  two  gears  must  remain 
in  a  constant  position  in  order  that  the  velocity  ratio  shall  bo 
constant.  The  essential  condition  for  constant  velocity  ratio  is, 
therefore,  that  the  position  of  the  centro  of  relative  motion  of  the 
gears  shall  remain  unchanged.  If  A  and  B,  Fig.  199,  are  tooth 
surfaces  in  contact  at  a,  their  only  possible  relative  motion,  if 
they  remain  in  contact,  is  slipping  motion  along  the  tangent  CD. 
The  centro  of  this  motion  must  be  in  EF,  a  normal  to  the  tooth 
surfaces  at  the  point  of  contact.  If  these  be  supposed  to  be 
teeth  of  a  pair  of  gears,  b  and  c,  whose  required  velocity  ratio 
is  known,  and  whose  centro,  be,  is  therefore  located,  then  in 
order  that  the  motion  communicated  from  one  gear  to  the  other 
through  the  point  of  contact,  a,  shall  be  the  required  motion,  it 
is  necessary  that  the  centro  of  the  relative  motion  of  the  teeth 
shall  coincide  with  be. 

198.  Illustration. — In  Fig.  200,  let  ac  and  ab  be  centers  of 
rotation  of  bodies  b  and  c,  and  the  required  velocity  ratio  is  such 
that  the  centro  of  b  and  c  falls  at  be.     Contact  between  b  and  c 
is  at  p.     The  only  possible  relative  motion  if  these  surfaces  re- 
main in  contact  is  slipping  along  CD;   hence  the  centro  of  this 
motion  must  be  on  EF,  the  normal  to  the  tooth  surfaces  at  the 
point  of  contact.     But  it  must  also  be  on  the  same  straight  line 


TOOTHED   WHEELS   OR   GEARS. 


351 


with  ac  and  ab\  hence  it  is  at  be,  and  the  motion  transmitted  for 
the  instant,  at  the  point  p,  is  the  required  motion,  because  its 
centro  is  at  be.  But  the  curves  touching  at  p  might  be  of  such 
form  that  their  common  normal  at  p  would  intersect  the  line  of 
centers  at  some  other  point,  as  K,  which  would  then  become  the 
centro  of  the  motion  of  b  and  c  for  the  instant,  and  would  corre- 
spond to  the  transmission  of  a  different  motion.  The  essential 
condition  to  be  fulfilled  by  tooth  outlines,  in  order  that  a  con- 
stant velocity  ratio  may  be  maintained,  may  therefore  be  stated 
as  follows:  The  tooth  outlines  must  be  such  that  their  normal  at 
the  point  o]  contact  shall  always  pass  through  the  centro  corre- 
sponding to  the  required  velocity  ratio. 

FIG.  199. 


FIG.  200. 


199.  Given  Tooth  Outline  to  Find  Form  of  Engaging  Tooth. 

—  Having  given  any  curve  that  will  serve  for  a  tooth  outline 
in  one  gear,  the  corresponding  curve  may  be  found  in  the  other 
gear,  which  will  engage  with  the  given  curve  and  transmit  a 


constant   velocity   ratio.      Let   -  •   be   the    given    velocity    ratio. 


=  the  sum  of  the  radii  of  the  two  gears.  Draw  the  line 
of  centers  AB,  Fig.  201.  Let  P  be  the  "pitch  point,"  i.e.,  the 
point  of  contact  of  the  pitch  circles  or  the  centro  of  relative 
motion  of  the  two  gears.  To  the  right  from  P  lay  off  a  distance 
PB=m\  from  P  toward  the  left  lay  off  PA  =n.  A  and  B  will 
then  be  the  required  centers  of  the  wheels,  and  the  pitch  circles 


352 


MACHINE  DESIGN. 


may  be  drawn  through  P.  Let  abc  be  any  given  curve  on  the 
wheel  A.  It  is  required  to  find  the  curve  in  B  which  shall  engage 
with  abc  to  transmit  the  constant  velocity  ratio  required.  A 
normal  to  the  point  of  contact  must  pass  through  the  centre. 
If,  therefore,  any  point,  as  a,  be  taken  in  the  given  curve,  and  a 
normal  to  the  curve  at  that  point  be  drawn,  as  aa,  then  when  a 
is  the  point  of  contact,  a  will  coincide  with  P.  Also,  if  cf  is  a 
normal  to  the  curve  at  c,  then  7-  will  coincide  with  P  when  c  is 
the  point  of  contact  between  the  gears;  and  since  b  is  in  the  pitch 
line,  it  will  itself  coincide  with  P  when  it  is  the  point  of  contact. 

FIG.  201. 

0? 


FlG.    202. 

Since  the  two  pitch  circles  must  roll  upon  each  other  without 
slipping,  it  follows  that  the  arc  Pa' =  arc  Pa,  arc  P6'=arc  Pb, 
and  arc  P^=arc  Pf. 

Rotate  the  point  a,  about  A,  through  the  angle  6.  At  the 
same  time  a'  rotates  backward  about  B  through  the  angle  6' 
and  a  and  a'  coincide  at  P.  Pa"  represents  the  rotated  position 
of  the  normal  aa.  Rotate  Pa"  about  B  through  the  angle  61 ' ; 
P  will  coincide  with  a'  and  a"  will  locate  the  point  a'  of  the  desired 
tooth  outline  of  gea-r  B.  The  point  bf  of  the  desired  outline  is 
readily  located  by  merely  laying  off  arc  PV  =arc  Pb. 

cf  is  located  by  the  same  method  we  employed  to  determine 
a1 '.  This  will  give  three  points  in  the  required  curve,  and  through 


TOOTHED   WHEELS  OR   GE4RS.  353 

these  the  curve  may  be  drawn.     The  curve  could,  of  course,  be 
more  accurately  determined  by  using  more  points. 

Many  curves  could  be  drawn  that  would  not  serve  for  tooth 
outlines;  but,  given  any  curve  that  will  serve,  the  corresponding 
curve  may  be  found.  There  would  be,  therefore,  almost  an 
infinite  number  of  curves  that  would  fulfill  the  requirements  of 
correct  tooth  outlines.  But  in  practice  two  kinds  of  curves  are 
found  so  convenient  that  they  are  most  commonly,  though  not 
exclusively,  used.  They  are  cycloidal  and  involute  curves. 

200.  Cycloidal  Tooth  Outlines. — It  is  assumed  that  the  char- 
acter of  cycloidal  curves  and  method  of  drawing  them  is  under- 
stood. 

In  Fig.  20?,  let  b  and  c  be  the  pitch  circles  of  a  pair  of  wheels, 
always  in  contact  at  be.  Also,  let  m  be  the  describing  circle  in 
contact  with  both  at  the  same  point.  M  is  the  describing  point. 
When  one  curve  rolls  upon  another,  the  centre  of  their  relative 
motion  is  always  their  point  of  contact.  For,  since  the  motion 
of  rolling  excludes  slipping,  the  two  bodies  must  be  stationary, 
relative  to  each  other,  at  their  point  of  contact;  and  bodies  that 
move  relative  to  each  other  can  have  but  one  such  stationary- 
point  in  common — their  centro.  When,  therefore,  m  rolls  in 
or  upon  b  or  c,  its  centro  relatively  to  either  is  their  point  of  con- 
tact. The  point  M,  therefore,  must  describe  curves  whose 
direction  at  any  point  is  at  right  angles  to  a  line  joining  that 
point  to  the  point  of  contact  of  m  with  the  pitch  circles.  Suppose 
the  two  circles  b  and  c  to  revolve  about  their  centers,  being  always 
in  contact  at  bc\  suppose  m  to  rotate  at  the  same  time  about  its 
center,  the  three  circles  being  always  in  contact  at  one  point  and 
having  no  slip.  The  point  M  will  then  describe  simultaneously 
a  curve,  ft7,  on  the  plane  of  b,  and  a  curve,  c',  on  the  plane  of  c. 
Since  M  describes  the  curves  simultaneously,  it  will  always  be 
the  point  of  contact  between  them  in  any  position.  And  since 
the  point  M  moves  always  at  right  angles  to  a  line  which  joins  it 


354  MACHINE  DESIGN. 

to  be,  therefore  the  normal  to  the  tooth  surfaces  at  their  point 
of  contact  will  always  pass  through  be,  and  the  condition  for 
constant  velocity  ratio  transmission  is  fulfilled.  But  these  curves 
are  precisely  the  epicycloid  and  hypocycloid  that  would  be  drawn 
by  the  point  M  in  the  generating  circle,  by  rolling  on  the  out- 
side of  b  and  inside  of  c.  Obviously,  then,  the  epicycloids  and 
hypocycloids  generated  in  this  way,  used  as  tooth  profiles,  will 
transmit  a  constant  velocity  ratio. 

This  proof  is  independent  of  the  size  of  the  generating  circle, 
and  its  diameter  may  therefore  equal  the  radius  of  c.  Then  the 
hypocycloids  generated  by  rolling  within  c  would  be  straight 
lines  coinciding  with  the  radius  of  c.  In  this  case  the  flanks 
of  the  teeth  of  c  become  radial  lines,  and  therefore  the  teeth  are 
thinner  at  the  base  than  at  the  pitch  line;  for  this  reason  they 
are  weaker  than  if  a  smaller  generating  circle  had  been  used.  All 
tooth  curves  generated  with  the  same  generating  circle  will  work 
together,  the  pitch  being  the  same.  It  is  therefore  necessary 
to  use  the  same  generating  circle  for  a  set  of  gears  which  need 
to  interchange* 

The  describing  circle  may  be  made  still  larger.  In  the  first 
case  the  curves  described  have  their  convexity  in  the  same  direction, 
i.e.,  they  lie  on  the  same  side  of  a  common  tangent.  When  the 
diameter  of  the  describing  circle  is  made  equal  to  the  radius 
of  c,  one  curve  becomes  a  straight-line  tangent  to  the  other  curve. 
As  the  describing  circle  becomes  still  larger,  the  curves  have  their 
convexity  in  opposite  directions.  As  the  circle  approximates 
equality  with  c,  the  hypocycloid  in  c  grows  shorter,  and  finally 
when  the  describing  circle  equals  c,  it  becomes  a  point  which  is 
the  generating  point  in  c,  which  is  now  the  generating  circle.  If 
this  point  could  be  replaced  by  a  pin  having  no  sensible  diameter, 
it  would  engage  with  the  epicycloid  generated  by  it  in  the  other 
gear  to  transmit  a  constant  velocity  ratio.  But  a  pin  without 

*  See  §  205. 


TOOTHED   WHEELS  OR    GEARS.  355 

sensible  diameter  will  not  serve  as  a  wheel-tooth,  and  a  proper 
diameter  must  be  assumed,  and  a  new  curve  laid  off  to  engage 
with  it  in  the  other  gear.     In  Fig.  203,  AB  is  the  epicycloid  gener- 
ated by  a  point  in  the  circumference  of  the 
other   pitch   circle.      CD   is   the    new  curve 
drawn  tangent  to  a  series  of  positions  of  the 
1  in  as   shown.      The   pin   will    engage   with 
this   curve,  CD,   and   transmit   the   constant 
velocity  ratio  as  required.     In  Fig.  202,  let  it  FlG 

be  supposed  that  when  the  three  circles  rotate 
constantly  tangent  to  each  other  at  tlie  pitch  point  be,  a  pencil  is 
fastened  at  the  point  M  in  the  circumference  of  the  describing 
circle.  If  this  pencil  be  supposed  to  mark  simultaneously  upon 
the  planes  of  b,  c,  and  that  of  the  paper,  it  will  describe  upon  b 
an  epicycloid,  on  c  a  hypocycloid,  and  on  the  plane  of  the 
paper  an  arc  of  the  describing  circle.  Since  M  is  always 
the  point  of  contact  of  the  cycloidal  curves  (because  it  gen- 
erates them  simultaneously),  therefore,  in  cycloidal  gear-teeth, 
the  locus  or  path  of  the  point  of  contact  is  an  arc  of  the  describ- 
ing circle.  The  ends  of  this  path  in  any  given  case  are  located 
by  the  points  at  which  the  addendum  circles  cut  the  describing 
circles. 

In  the  cases  already  considered,  where  an  epicycloid  in  one 
wheel  engages  with  a  hypocycloid  in  the  other,  the  contact  of 
the  teeth  with  each  other  is  all  on  one  side  of  the  line  of  centers. 
Thus,  in  Fig.  202,  if  the  motion  be  reversed,  the  curves  will  be 
in  contact  until  M  returns  to  be  along  the  arc  MD-bc\  but  after 
M  passes  be  Contact  will  cease.  If  c  were  the  driving-wheel,  the 
point  of  contact  would  approach  the  line  of  centers;  if  b  were  the 
driving-wheel  the  point  of  contact  would  recede  from  the  line  of 
centers.  Experience  shows  that  the  latter  gives  smoother  running 
because  of  better  conditions  as  regards  friction  between  the  tooth 
surfaces.  It  would  be  desirable,  therefore,  that  the  wheel  with 
the  epicycloidal  curves  should  always  be  the  driver.  But  it 


356  MACHINE  DESIGN. 

should  be  possible  to  use  either  wheel  as  driver  to  meet  the  varying 
conditions  in  practice. 

Another  reason  why  contact  should  not  be  all  on  one  side 
of  the  line  of  centers  may  be  explained  as  follows : 

201.  Definitions:  Pitch-arc,  Arc  of  Action,  Line  of  Pressure. 
—The  angle  through  which  a  gear-wheel  turns  while  one  of  its 
teeth  is  in  contact  with  the  corresponding  tooth  in  the  other  gear 
is  called  the  angle  of  action.  It  is  found  by  drawing  radial  lines 
from  the  center  to  the  pitch  circle  at  the  two  tooth  positions  cor- 
responding to  the  beginning  and  end  of  engagement.  The  arc  of 
the  pitch  circle  corresponding  to  the  angle  of  action  is  called  the 
fire  of  action. 

The  arc  of  action  must  be  greater  than  the  "  pitch  arc  "  (the  arc 
of  the  pitch  circle  that  includes  one  tooth  and  one  space),  or  else 
contact  will  cease  between  one  pair  of  teeth  before  it  begins  between 
the  next  pair.  Constrainment  would  therefore  not  be  complete, 
and  irregular  velocity  ratio  with  noisy  action  would  result. 

In  Fig.  204,  let  AB  and  CD  be  the  pitch  circles  of  a  pair  of 
gears  and  E  the  describing  circle.  Let  an  arc  of  action  be  laid 
off  on  each  of  the  circles  from  P,  as  Pa,  PC,  and  Pe.  Through 
e,  about  the  center  O,  draw  an  addendum  circle;  i.e.,  the  circle 
which  limits  the  points  of  the  teeth.  Since  the  circle  E  is  the 
path  of  the  point  of  contact,  and  since  the  addendum  circle 
limits  the  points  of  the  teeth,  their  intersection,  e,  is  the  point  at 
which  contact  ceases,  rotation  being  as  indicated  by  the  arrow. 
If  the  pitch  arc  just  equals  the  assumed  arc  of  action,  contact 
will  be  just  beginning  at  P  when  it  ceases  at  e\  bnt  if  the  pitch 
arc  be  greater  than  the  arc  of  action,  contact  will  not  bsgin  at 
P  till  after  it  has  ceased  at  e,  and  there  will  be  an  interval  when 
AB  will  not  drive  CD.  The  greater  the  arc  of  action  the  greater 
the  distance  of  e  from  P  on  the  circumference  of  the  describing 
circle.  The  direction  of  pressure  between  the  teeth  is  always 
a  normal  to  the  tooth  surface,  and  this  always  passes  through 
xhe  pitch  point;  therefore  the  greater  the  arc  of  action — i.e.. 


TOOTHED   WHEELS   OR  GE/tRS. 


357 


the  greater  the  distance  of  e  from  P— the  greater  the  obliquity 
of  the  line  of  pressure.  The  pressure  may  be  resolved  into  two 
components,  one  at  right  angles  to  the  line  of  centers  and  the 
other  parallel  to  it.  The  first  is  resisted  by  the  teeth  of  the  follower- 
wheel,  and  is  effective  to  produce  the  desired  rotation,  while  the 
second  tends  to  crowd  the  journals  apart,  and  therefore  produces 
pressure  with  resulting  friction.  Hence  it  follows  that  the  greater 
the  arc  of  action  the  greater  will  be  the  average  obliquity  of 
the  line  of  pressure,  and  therefore  the  greater  the  component 


FIG.  204. 


FIG.  205. 


of  the  pressure  that  produces  wasteful  friction.  If  it  can  be 
arranged  so  that  the  arc  of  action  shall  be  partly  on  each 
side  of  the  line  of  centers,  the  arc  of  action  may  be  made 
greater  than  the  pitch  arc  (usually  equal  to  about  i-J  times 
the  pitch  arc);  then  the  obliquity  of  the  pressure-line  may 
be  kept  within  reasonable  limits,  contact  between  the  teeth 
will  be  insured  in  all  positions,  and  either  wheel  may  be  the 
driver.  This  is  accomplished  by  using  two  describing  circles  as 
in  Fig.  205.  Suppose  the  four  circles  A,  B,  a,  and  /?  to  roll  con^ 
stantly  tangent  at  P.  a  will  describe  an  epicycloid  on  the  plane 
of  By  and  a  hypocycloid  on  the  plane  of  A.  These  curves  will 
engage  with  each  other  to  drive  correctly.  /?  will  describe  an 
epicycloid  on  A,  and  a  hypocycloid  on  B.  These  curves  will 
engage,  also,  to  drive  correctly.  If  the  epicycloid  and  hypocycloid 
in  each  gear  be  drawn  through  the  same  point  on  the  pitch  circle, 
a  double  curve  tooth  outline  will  be  located,  and  one  curve  will 


358  MACHINE  DESIGN. 

engage  on  one  side  of  the  line  of  centers  and  the  other  on  the 
other  side.  If  A  drives  as  indicated  by  the  arrow,  contact  will 
begin  at  D,  and  the  point  of  contact  will  follow  an  arc  of  a  to  P, 
and  then  an  arc  of  /?  to  C. 

202.  Involute  Tooth  Outlines. — If  a  string  is  wound  around  a 
cylinder  and  a  pencil-point  attached  to  its  end,  thi  s  point  will 
trace  an  involute  on  a  plane  normal  to  the  axis  of  the  cylinder 
as  the  string  is  unwound  from  the  cylinder.  Or,  if  the  point 
be  constrained  to  follow  a  tangent  to  the  cylinder,  and  the  string 
be  unwound  by  rotating  the  cylinder  about  its  axis,  the  point 
will  trace  an  involute  on  a  plane  that  rotates  with  the  cylinder. 

Illustration. — Let  a,  Fig.  206,  be  a  circular  piece  of  wood  free 
to  rotate  about  C;  /?  is  a  circular  piece  of  cardboard  made  fast 
to  a]  AB  is  a  straight-edge  held  on  the  circumference  of  a, 
having  a  pencil-point  at  B.  As  B  moves  along  the  straight-edge 
to  A,  a  and  /?  rotate  about  C,  and  B  traces  an  involute  DB  upon 
/?,  the  relative  motion  of  the  tracing  point  and  /?  being  just  the 
same  as  if  the  string  had  been  simply  unwound  from  a  fixed. 
If  the  tracing  point  is  caused  to  return  along  the  straight-edge 
it  will  trace  the  involute  BD  in  a  reverse  direction. 


FIG.  206.  FIG.  207. 

The  centre  of  the  tracing  point  is  always  the  point  of  tan- 
gen  cy  of  the  string  with  the  cylinder;  therefore  the  string,  or 
straight-edge,  in  Fig.  208,  is  always  at  right  angles  to  the  direc- 
tion of  motion  of  the  tracing  point,  and  hence  is  always  a  normal 
to  the  involute  curve.  Let  a  and  /?,  Fig.  207,  be  two  base  cylin- 
ders; let  AB  be  a  cord  wound  upon  a  and  /?  and  passing  through 
the  centro  P,  which  corresponds  to  the  required  velocity  ratio. 


TOOTHED   WHEELS  OR   GEARS.  359 

Let  a  and  /?  be  supposed  to  rotate  so  that  the  cord  is  wound  from 
/?  upon  a.  Then  any  point  in  the  cord  will  move  from  A  toward 
B,  and,  if  it  be  a  tracing-point,  will  trace  an  involute  of  a  on  the 
plane  of  a  (extended  beyond  the  base  cylinder) ,  and  will  also  trace 
an  involute  of  /?  upon  the  plane  of  /?.  These  two  involutes  will 
serve  for  tooth  profiles  for  the  transmission  of  the  required  con- 
stant velocity  ratio,  because  AB  is  the  constant  normal  to  both 
curves  at  their  point  of  contact,  and  it  passes  through  P,  the 
centro  corresponding  to  the  required  velocity  ratio.  Hence  the 
necessary  condition  is  fulfilled.  The  pitch  circles  will  have  OP 
and  O'P  as  their  respective  radii. 

Since  a  point  in  the  line  AB  describes  the  two  involute  curves 
simultaneously,  the  point  of  contact  of  the  curves  is  always  in 
the  line  AB.  And  hence  AB  is  the  path  of  the  point  of  contact. 
In  any  given  case  the  two  ends  of  the  path  lie  at  the  intersections 
of  the  addendum  circles  with  AB.  Should  either  addendum 
circle  intersect  the  line  of  action  outside  of  the  portion  lying 
between  A  and  B,  "  interference  "  takes  place.  In  such  cases 
the  addendum  may  be  shortened  or  the  profile  of  the  tooth- 
point  modified  from  the  true  involute. 

To  avoid  interference,  the  allowable  length  of  addendum 
in  any  case  (conversely,  the  least  number  of  teeth  of  pinion) 
may  be  computed  by  Bach's  formula: 


27T 


02  =  addendum  of  gear- tooth  (commonly  -  =  —  ),  inches; 

\  P       7T  / 

NI  =  number  of  teeth  of  pinion; 
N2  =  number  of  teeth  of  gear  (=  oo  for  rack); 
a  =  angle  between  line  of  action  and  line  of  centers; 
P  =  circular  pitch,  inches. 

One  of  the  advantages  of  involute  curves  for  tooth  profiles 
is  that  a  change  in  distance  between  centers  of  the  gears  does 
not  interfere  with  the  transmission  of  a  constant  velocity  ratio. 


360  MACHINE  DESIGN. 

This  may  be  proved  as  follows :  In  Fig.  207,  from  similar  triangles 
O  7?       OP 
TyX^rxp'   tnat  'ls>  tne  ratio  of  the  radii  of  the  base  circles  (i.e., 

sections  of  the  base  cylinders)  is  equal  to  the  ratio  of  the  radii  of 
the  pitch  circles.  This  ratio  equals  the  inverse  ratio  of  angular 
velocities  of  the  gears.  Suppose  now  that  O  and  O'  be  moved 
nearer  together;  the  pitch  circles  will  be  smaller,  but  the  ratio 
of  their  radii  must  be  equal  to  the  unchanged  ratio  of  the 
radii  of  the  base  circles,  and  therefore  the  velocity  ratio  remains 
unchanged.  Also  the  involute  curves,  since  they  are  generated 
from  the  same  base  cylinders,  will  be  the  same  as  before,  and 
therefore,  with  the  same  tooth  outlines,  the  same  constant  velocity 
ratio  will  be  transmitted  as  before. 

203.  Racks. — A  rack  is  a  wheel  whose  pitch  radius  is  infinite; 
its  pitch  circle,  therefore,  becomes  a  straight  line,  and  is  tangent 
to  the  pitch  circle  of  the  wheel,  or  pinion,*  with  which  the  rack 
engages.  The  line  of  centers  is  a  normal  to  the  pitch  line  of  the 
rack,  through  the  center  of  the  pitch  circle  of  the  pinion.  The 
pitch  of  the  rack  is  determined  by  laying  off  the  circular  pitch 
of  the  engaging  wheel  on  the  pitch  line  of  the  rack.  The  curves 
of  the  cycloidal  rack- teeth,  like  those  of  wheels  of  finite  radius, 
may  be  generated  by  a  point  in  the  circumference  of  a  circle  which 
rolls  on  the  pitch  circle.  Since,  however,  the  pitch  circle  is  now 
a  straight  line,  the  tooth  curves  will  be  cycloids,  both  for  flanks 
and  faces.  In  Fig.  208,  AB  is  the  pitch  circle  of  the  pinion  and 
CD  is  the  pitch  line  of  the  rack;  a  and  b  are  describing  circles. 
Suppose,  as  before,  that  all  move  without  slipping  and  are  con- 
stantly tangent  at  P.  A  point  in  the  circumference  of  a  will  then 
describe  simultaneously  a  cycloid  on  CD,  and  a  hypocycloid 
within  AB.  These  will  be  correct  tooth  outlines.  Also,  a  point 
in  the  circumference  of  b  will  describe  a  cycloid  on  CD,  and  an 
epicycloid  on  AB.  These  will  be  correct  tooth  outlines.  To 

*  Pinion  is  a  word  to  denote  a  gear  having  a  low  number  of  teeth,  or  the  smaller 
one  of  a  pair  of  engaging  gears. 


TOOTHED    WHEELS  OR   GEARS.  361 

find  the  path  of  the  point  of  contact,  draw  the  addendum  circle 
EF  of  the  pinion,  and  the  addendum  line  GH  of  the  rack.  When 
the  pinion  turns  clockwise  and  drives 
the  rack,  contact  will  begin  at  e  and 


.F 

FIG.  208.  FIG.  209. 

follow  arcs  of  the  describing  circles  through  P  to  K.  It  is  ob- 
vious that  a  rack  cannot  be  used  where  rotation  is  continuous  in 
one  direction,  but  only  where  motion  is  reversed. 

Involute  curves  may  also  be  used  for  the  outlines  of  rack 
teeth.  Let  CD  and  CD',  Fig.  209,  be  the  pitch  lines.  When  it 
is  required  to  generate  involute  curves  for  tooth  outlines,  for  a 
pair  of  gears  of  finite  radius,  a  line  is  drawn  through  the  pitch 
point  at  a  given  angle  to  the  line  of  centers  (usually  75°) ;  this 
line  is  the  path  of  the  point  which  generates  two  involutes  simul- 
taneously, and  therefore  the  path  of  the  point  of  contact  between 
the  tooth  curves.  It  is  also  the  common  tangent  to  the  two  base 
circles,  which  may  now  be  drawn  and  used  for  the  describing  of 
the  involutes.  To  apply  this  to  the  case  of  a  rack  and  pinion, 
draw  EF,  Fig.  209,  making  the  desired  angle  with  the  line  of 
centers,  OP.  The  base  circles  must  be  drawn  tangent  to  this 
line;  AB  will  therefore  be  the  base  circle  for  the  pinion.  But 
the  base  circle  in  the  rack  has  ah  infinite  radius,  and  a  circle  of 
infinite  radius  drawn  tangent  to  EF  would  be  a  straight  line 
coincident  with  EF.  Therefore  EF  is  the  base  line  of  the  rack. 
But  an  involute  to  a  base  circle  of  infinite  radius  is  a  straight 
line  normal  to  the  circumference — in  this  case  a  straight  line  per- 
pendicular to  EF.  Therefore  the  tooth  profiles  of  a  rack  in  the 
involute  system  will  always  be  straight  lines  perpendicular  to  the 
path  of  the  describing  point,  and  passing  through  the  pitch  points. 
If,  in  Fig.  209,  the  pinion  move  clockwise  and  drive  the  rack,  the 


362 


MACHINE  DESIGN. 


contact  will  begin  at  E,  the  intersection  of  the  addendum  line 
of  the  rack  GH,  and  the  path  of  the  point  of  contact  EF,  and 
will  follow  the  line  EF  through  P  to  the  point  where  EF  cuts  the 
addendum  circle  LM  of  the  pinion. 

204.  Annular  Gears. — Both  centers  of  a  pair  of  gears  may  be 
on  the  same  side  of  the  pitch  point.  This  arrangement  corre- 
sponds to  what  is  known  as  an  annular  gear  and  pinion.  Thus, 
in  Fig.  210,  AB  and  CD  are  the  pitch  circles,  and  their  centers 
are  both  above  the  pitch  point  P, 
Teeth  may  be  constructed  to  trans- 


FlG.  2IO. 


FIG.  211. 


mit  rotation  between  AB  and  CD.  AB  will  be  an  ordinary 
spur  pinion,  but  it  is  obvious  that  CD  becomes  a  ring  of  metal 
with  teeth  on  the  inside,  i.e.,  it  is  an  annular  gear.  In  this  case 
a.  and  /?  may  be  describing  circles  for  cycloidal  teeth,  and  a  point 
in  the  circumference  of  a  will  describe  hypocycloids  simultaneously 
on  the  planes  of  AB  and  CD\  and  a  point  in  the  circumference 
of  /?  will  describe  epicycloids  simultaneously  on  the  planes  of  AB 
and  CD.  These  will  engage  to  transmit  a  constant  velocity 
ratio.  Obviously  the  space  inside  of  an  annular  gear  corresponds 
to  a  spur- gear  of  the  same  pitch  and  pitch  diameter,  with  tooth 
curves  drawn  with  the  same  describing  circle.  Let  EF  and  GH, 
Fig.  210,  be  the  addendum  circles.  If  the  pinion  move  clockwise 
driving  the  annular  gear,  the  path  of  the  point  of  contact  will  be 
from  e  along  the  circumference  of  a  to  P,  and  from  P  along  the 
circumference  of  /?  to  K. 

The  construction  of  involute  teeth  for  an  annular  gear  and 
pinion  involves  exactly  the  same  principles  as  in  the  case  of  a 


TOOTHED   WHEELS   OR    GEARS.  363 

pair  of  spur-gears.  The  only  difference  of  detail  is  that  the 
describing  point  is  in  the  tangent  to  the  base  circles  produced 
instead  of  being  between  the  points  of  tangency.  Let  O  and  O\ 
Fig.  211,  be  the  centers,  and  AB  and  //  the  pitch  circles  of  an 
annular  gear  and  pinion.  Through  P,  the  point  of  tangency  of 
the  pitch  circles,  draw  the  path  of  the  point  of  contact  at  the  given 
angle  with  the  line  of  centers.  With  O  and  O'  as  centers  draw 
tangent  circles  to  this  line.  These  will  be  the  involute  base 
circles.  Let  the  tangent  be  replaced  by  a  cord,  made  fast,  say, 
at  K',  winding  on  the  circumference  of  the  base  circle  CK',  to  D, 
and  then  around  the  base  circle  FE  in  the  direction  of  the  arrow, 
and  passing  over  the  pulley  G,  which  holds  it  in  line  with  PB. 
If  rotation  be  supposed  to  occur  with  the  two  pitch  circles  always 
tangent  at  P  without  slipping,  any  point  in  the  cord  beyond  P 
toward  G  will  describe  an  involute  on  the  plane  //,  and  another 
on  the  plane  of  AB.  These  will  be  the  correct  involute  tooth 
profiles  required.  Draw  NQ  and  LM,  the  addendum  circles. 
Then  if  the  pinion  move  clockwise,  driving  the  annular  gear, 
the  point  of  contact  starts  from  e  and  moves  along  the  line  GH 
through  P  to  K. 

When  a  pair  of  spur-gears  mesh  with  each  other,  the  direction 
of  rotation  is  reversed.  But  an  annular  gear  and  pinion  meshing 
together  rotate  in  the  same  direction. 

205.  Interchangeable  Sets  of  Gears. — In  practice  it  is  desirable 
to  have  "interchangeable  sets  "  of  gears;  i.e.,  sets  in  which  any 
gear  will  "mesh  "  correctly  with  any  other,  from  the  smallest 
pinion  to  the  rack,  and  in  which,  except  for  limiting  conditions 
of  size,  any  spur-gear  will  mesh  with  any  annular  gear.  Inter- 
changeable sets  may  be  made  in  either  the  cycloidal  or  involute 
system.  A  necessary  condition  in  any  set  is  that  the  pitch  shall 
be  constant,  because  the  thickness  of  tooth  on  the  pitch  line  must 
always  equal  the  width  of  the  space  (less  backlash).  If  this 
condition  is  unfulfilled  they  cannot  engage,  whatever  the  form 
of  the  tooth  outlines. 


364  MACHINE  DESIGN. 

The  second  condition  for  an  interchangeable  set  in  the  cycloL 
dal  system  is  that  the  size  of  the  describing  circle  shall  be  constant. 
If  the  diameter  of  the  describing  circle  equals  the  radius  of  the 
smallest  pinion's  pitch  circle,  the  flanks  of  this  pinion's  teeth  will 
be  radial  lines,  and  the  tooth  will  therefore  be  thinner  at  the  base 
than  at  the  pitch  line.  As  the  gears  increase  in  size  with  this 
constant  size  of  describing  circle,  the  teeth  grow  thicker  at  the 
base;  hence  the  weakest  teeth  are  those  of  the  smallest  pinion. 

It  is  found  unadvisable  to  make  a  pinion  with  less  than  twelve 
teeth.  If  the  radius  of  a  fifteen -tooth  pinion  be  selected  for  the 
diameter  of  the  describing  circle,, the  flanks  which  bound  a  space 
in  a  twelve-tooth  pinion  will  be  very  nearly  parallel,  and  may 
therefore  be  cut  with  a  milling-cutter.  This  would  not  be  possi- 
ble if  the  describing  circle  were  made  larger,  causing  the  space 
to  become  wider  at  the  bottom  than  at  the  pitch  circle.  There- 
fore the  maximum  describing  circle  for  milled  gears  is  one  whose 
diameter  equals  the  pitch  radius  of  a  fifteen-tooth  pinion  and 
it  is  the  one  usually  selected.  Each  change  in  the  number  of 
teeth  with  constant  pitch  causes  a  change  in  the  size  of  the  pitch 
circle.  Hence  the  form  of  the  tooth  outline,  generated  by  a 
describing  circle  of  constant  diameter,  also  changes.  For  any 
pitch,  therefore,  a  separate  cutter  would  be  required  corresponding 
to  every  number  of  teeth,  to  insure  absolute  accuracy.  Practi- 
cally, however,  this  is  not  necessary.  The  change  in  the  form 
of  tooth  outline  is  much  greater  in  a  small  gear,  for  any  increase 
in  the  number  of  teeth,  than  in  a  large  one.  It  is  found  that 
twenty-four  cutters  will  cut  all  possible  gears  of  the  same  pitch 
with  sufficient  practical  accuracy.  The  range  of  these  cutters 
is  indicated  in  the  following  table,  taken  from  Brown  and  Sharpe's 
" Treatise  on  Gearing." 

These  same  principles  of  interchangeable  sets  of  gears  with  cy- 
cloldal  tooth  outlines  apply  not  only  to  small  milled  gears  as  above, 
but  also  to  large  cast  gears  with  tooled  or  untooled  tooth  surfaces. 


TOOTHED   WHEELS    OK   GEARS 


365 


TABLE  XXXI. 


Cutter  A  cuts  12  teeth 


Cutter  M  cuts    27  to    29  teeth 


B     " 

i3 

" 

C     " 

14 

1C 

D    " 

i5 

" 

E     " 

16 

" 

F     " 

ij 

•* 

G    " 

18 

H 

H    " 

X9 

" 

I      " 

20 

11 

J     " 

21 

to  22  teeth 

K    " 

23 

"24     " 

L    " 

25 

"  26     " 

N  " 

30" 

33 

0  " 

34" 

37 

P  " 

38" 

42 

Q  " 

43  " 

49 

R  " 

5°  " 

59 

S   " 

60  " 

74 

T  " 

75" 

99 

U  " 

100  " 

149 

V  " 

150  " 

249 

W  " 

250" 

rack 

X  " 

rack 

206.  Interchangeable  Involute  Gears. — In  the  involute  system 
the  second  condition  of  interchangeability  is  that  the  angle  between 
the  common  tangent  to  the  base  circles  and  the  line  of  centers  shall 
be  constant.  This  may  be  shown  as  follows:  Draw  the  line  of 
centers,  AB,  Fig.  212.  Through  P,  the  assumed  pitch  point, 
draw  CD,  and  let  it  be  the  constant  common  tangent  to  all  base 


circles  from  which  involute  tooth  curves  are  to  be  drawn.  Draw 
any  pair  of  pitch  circles  tangent  at  P,  with  their  centers  in  the 
line  AB.  About  these  centers  draw  circles  tangent  to  CD;  these 
are  base  circles,  and  CD  may  represent  a  cord  that  winds  from 
one  upon  the  other.  A  point  in  this  cord  will  generate  simul- 
taneously involutes  that  will  engage  for  the  transmission  of  a  con- 
stant velocity  ratio.  But  this  is  true  of  any  pair  of  circles  that 
have  their  centers  in  AB,  and  are  tangent  to  CD.  Therefore, 


366 


MACHINE  DESIGN. 


if  the  pitch  is  constant,  any  pair  of  gears  that  have  the  base  circles 
tangent  to  the  line  CD  will  mesh  together  properly.  As  in  the 
cycloidal  gears,  the  involute  tooth  curves  vary  with  a  variation 
in  the  number  of  teeth,  and,  for  absolute  theoretical  accuracy, 
there  would  be  required  for  each  pitch  as  many  cutters  as  there 
are  gears  with  different  numbers  of  teeth.  The  variation  is 
least  at  the  pitch  line,  and  increases  with  the  distance  from  it. 
The  involute  teeth  are  usually  used  for  the  finer  pitches  and  the 
cycloidal  teeth  for  the  coarser  pitches ;  and  since  the  amount  that 
the  tooth  surface  extends  beyond  the  pitch  line  increases  with  the 
pitch,  it  follows  that  the  variation  in  form  of  tooth  curves  is 
greater  in  the  coarse  pitch  cycloidal  gears  than  in  the  fine  pitch 
involute  gears.  For  this  reason,  with  involute  gears,  it  is  only 
necessary  to  use  eight  cutters  for  each  pitch.  The  range  is  shown 
in  the  following  table,  which  is  also  taken  from  Brown  and  Sharpe's 
"Treatise  on  Gearing": 

TABLE  XXXII. 

No.  i  will  cut  wheels  from  135  teeth  to  racks 
2 

3 

4 

s 

6 

7 

8 

207.  Laying  Out  Gear- teeth.  —  Exact  and  Approximate 
Methods. — There  is  ordinarily  no  reason  why  an  exact  method 
for  laying  out  cycloidal  or  involute  curves  for  tooth  outlines  should 
not  be  used,  either  for  large  gears  or  gear  patterns,  or  in  making 
drawings.  It  is  required  to  lay  out  a  cycloidal  gear.  The  pitch, 
and  diameters  of  pitch  circle  and  describing  circle  are  given. 

Draw  the  pitch  circle  on  the  drawing-paper,  using  a  fine  line. 
On  a  flat  piece  of  tracing-cloth  or  thin,  transparent  celluloid  draw 
a  circle  the  size  of  the  generating  circle.  Use  a  fine,  clear  line. 


« 

55 

'     "134  inclusive 

' 

35 

'      "     54         " 

c 

26 

'      "     34        " 

< 

21 

'      "     25         " 

( 

17 

'        "       20 

' 

14 

'     "     16 

f 

12 

:    "    13 

TOOTHED    WHEELS   OR   GEARS. 


367 


Place  it  over  the  drawn  pitch  circle  so  that  it  is  tangent  to  the 
latter  at  P  as  shown  in  Fig.  213.  AB  is  an  arc  of  the  pitch  circle. 
Insert  a  needle  point  at  P,  and  using  it  as  a  pivot  swing  the 
tracing-cloth  in  the  direction  of  the  arrow  a  very  short  distance, 
so  that  the  generating  circle  cuts  the  pitch  circle  at  a  new  point 
Q,  as  shown  exaggeratedly  in  Fig.  214.  Q  should  be  taken  very  close 


FIG.  213.  FIG.  214.  FIG.  215.  FIG.  216. 

to  P.  Insert  a  needle  point  at  Q,  remove  the  one  at  P,  and  swing 
the  cloth,  about  Q  as  a  pivot,  in  the  direction  of  the  arrow  until 
the  two  circles  are  tangent  at  Q.  (See  Fig.  215.)  The  point  P 
of  the  tracing- cloth  now  lies  off  the  pitch  circle  a  short  distance. 
With  a  needle  point  prick  its  present  position  through  to  the 
drawing-paper.  Now  with  Q  as  a  pivot  rotate  the  tracing-cloth 
until  the  two  circles  intersect  at  a  point  R  slightly  beyond  Q. 
Insert  needle  at  R  and  remove  the  one  at  Q.  Swing  tracing- 
cloth  about  R  until  R  becomes  the  point  of  tangency  of  the  two 
circles  and  then  prick  the  new  position  of  P  through  to  the  drawing- 
paper.  Taking  points  very  near  together  and  repeating  the 
operation  gives  a  close  approximation  to  true  rolling  of  the  gen- 
erating circle  on  the  pitch  circle  and  therefore  the  path  of  the 
point  P  as  marked  on  the  drawing-paper  by  pricked  points  is  an 
epicycloid  and  may  be  used  for  the  face  of  the  tooth. 

Next  place  the  tracing-cloth  on  the  inside  of  the  pitch  circle, 
as  shown  in  Fig.  216,  with  the  generating  circle  tangent  to  the 
pitch  circle  at  the  original  point  P.  Using  the  method  just 
described  to  prevent  slipping,  roll  the  generating  circle,  in  the 
direction  of  the  arrow,  on  the  pitch  circle  and  the  path  traced 
by  P  as  marked  by  pricked  points  on  the  drawing-paper  will  be 
a  hypocycloid  for  the  flank  of  the  tooth. 


368 


MACHINE  DESIGN. 


The  compound  curve  aPb,  Fig.  217,  has  now  been  traced, 
which  forms  the  basis  of  the  completed  tooth  outline. 

AB  is  an  arc  of  the  pitch  circle  whose  center  is  at  O.  With 
O  as  center,  swing  in  the  addendum  circle  'CD  and  the  full 
depth  circle  EF,  according  to  the  proportions  given  in  §  208. 
With  a  radius  equal  to  T^  of  the  circular  pitch  draw  the  fillet 
cd  tangent  to  EF  and  aPb.  The  completed  tooth  profile  is  the 
curve  cdPe. 

Cut  a  wooden  template  to  fit  the  tooth  curve,  and  make  it 
fast  to  a  wooden  arm  free  to  rotate  about  O,  making  the  edge  of 
the  template  coincide  with  cdPe.  It  may  now  be  swung  succes- 
sively to  the  other  pitch  points,  and  the  tooth  outline  may  be  drawn 
by  the  template  edge.  This  gives  one  side  of  all  of  the  teeth. 
The  arm  may  now  be  turned  over  and  the  other  sides  of  the 
teeth  may  be  drawn  similarly. 


\  \ 


J  J 


F          B         D 

FIG.  217. 


FIG.  218. 


It  is  required  to  lay  out  exact  involute  teeth.  The  pitch, 
pitch-circle  diameter,  and  angle  of  the  common  tangent  are 
given. — Draw  the  pitch  circle  AB,  Fig.  218,  and  the  line  of 
centers  OO1 '.  Through  the  pitch  point  P  draw  CD,  the  common 
tangent  to  the  base  circles,  making  the  angle  /?  with  the  line  of 
centers.  Draw  the  base  circle  EF  about  O  tangent  to  CD. 

On  a  piece  of  flat  tracing-cloth  draw  a  fine,  clear,  straight  line 
and  lay  the  tracing-cloth  over  the  drawing  so  that  this  line  coin- 
cides with  CD.  Take  a  needle  point  and  insert  it  at  the  point 


TOOTHED   WHEELS   OR   GEARS.  369 

of  tangency  Q.  With  another  needle,  mark  the  point  P  on  the 
tracing-cloth.  Now,  employing  a  pair  of  needle  points  to  pre- 
vent slipping,  roll  the  traced  line  on  the  base  circle  EF,  prick- 
ing the  path,  aPb,  of  the  point  P  of  the  tracing-cloth  through  to 
the  drawing-paper.  This  path  is  the  involute  of  the  base  circle 
and  is  the  basis  of  the  involute  tooth  outline.  To  complete  the 
latter  proceed  as  follows:  Draw  the  addendum  circle  GH  and 
the  full-depth  circle  JK.  In  general  JK  will  lie  inside  of  the 
base  circle  EF  and  it  will  be  necessary  to  extend  the  tooth  outline 
inward  beyond  a.  About  O  swing  a  circle  whose  diameter  equals 
one  half  the  circular  pitch  and  draw  ac  tangent  to  it.  With  a 
radius  equal  to  TV  of  the  circular  pitch  swing  in  the  fillet  de  tan- 
gent to  JK  and  ac.  deaPj  is  the  completed  outline.  If  the 
gear  has  20  teeth  or  less  ac  should  be  made  a  radial  line.  If 
EF  lies  inside  of  JK  we  draw  the  fillet  tangent  to  JK  and  aP* 

208.  Gear  Proportions. — The  following  formulas  and  table 
are  given  to  assist  in  the  practical  proportioning  of  gears  : 
Let  D  =  pitch  diameter; 
DI  =  outside  diameter; 

D%=  diameter  of  a  circle  through  the  bottom  of  spaces; 
P  =  circular  pitch  =  space  on  the  pitch  circle  occupied  by 

a  tooth  and  a  space ; 

p  =  diametral  pitch  =  number  of  teeth  per  inch  of  pitch- 
circle  diameter; 
N  =  number  of  teeth; 
/  =  thickness  of  tooth  on  pitch  line ; 
a  =  addendum; 

*  Approximate  tooth  outlines  may  be  drawn  by  the  use  of  instruments,  such 
as  the  Willis  odontograph,  which  locates  the  centers  of  approximate  circular  arcs; 
the  templet  odontograph,  invented  by  Prof.  S.  W.  Robinson;  or  by  some  geomet- 
rical or  tabular  method  for  the  location  of  the  centers  of  approximate  circular 
arcs.  For  descriptions,  see  "Elements  of  Mechanism,"  Willis;  "Kinematics," 
McCord;  "Teeth  of  Gears,"  George  B.  Grant;  "Treatise  on  Gearing,"  pub- 
lished by  Brown  and  Sharpe. 


370 


MACHINE  DESIGN. 


c  =  clearance; 

d  =  working  depth  of  spaces; 

d-i  =full  depth  of  spaces. 


Then 


AT  +  2 


;    D2=D-2(a+c)', 


Dx 
T' 


Dr.  PN 

P=jji    P  —  i    N-Dti 

_Nm     D_*L. 
=  -77;     P=-T]     t  =  — =~,  no  backlash; 

P  p  2         2p 


c= — = — =~r~i 

IO       2O       p2O 


a=—  inches. 
P 


The  following  dimensions  are  given  as  a  guide ;  they  may  be 
varied  as  conditions  of  design  require:  Width  of  face  =  about  $P\ 
thickness  of  rim  =  1.25/5  thickness  of  arms  =  i.25/,  no  taper. 
The  rim  may  be  reinforced  by  a  rib,  as  shown  in  Fig.  219.  Diam- 
eter of  hub=2Xdiameter  of  shaft.  Length  of  hub  =  width  of 
face  +  J";  width  of  arm  at  junction  with  hub=J  circumference 
of  the  hub  for  six  arms.  Make  arms  taper  about  |-"  per  foot  on 
each  side. 

TABLE  XXXIII. 


Diametral 
Pitch. 

Circular  Pitch. 

Thickness  of 
Tooth  on  the 
Pitch  Line. 

Diametral 
Pitch. 

Circular  Pitch. 

Thickness  of 
Tooth  on  the 
Pitch  Line. 

P 

P 

t 

P 

P 

t 

* 

6.283 

3-I4I 

3i 

.897 

•449 

1 

4.189 

2.094 

4 

.785 

•393 

I 

3-I4I 

I-57I 

5 

.628 

.314 

l£ 

1.256 

6 

•523 

.262 

J4 

2.094 

1.047 

7 

•449 

.224 

i| 

•795 

.897 

8 

•393 

.196 

2 

•571 

.785 

9 

•349 

.174 

2i 

•396 

.698 

10 

.314 

•  J57 

2* 

.256 

.628 

12 

.262 

•131 

2f 

.142 

•571 

14 

.224 

.  112 

3 

.047 

•523 

TOOTHED  WHEELS  OR  GEARS.  371 

209.  Strength  of  Gear-teeth. — The  maximum  work  trans- 
mitted by  a  shaft  per  unit  time  may  usually  be  accurately  estimated ; 
and,  if  the  rate  of  rotation  is  known,  the  torsional  moment  may 
be  found.  Let  O,  Fig.  220,  represent  the  axis  of  a  shaft  perpen- 
dicular to  the  paper.  Let  A  =  maximum  work  to  be  transmitted 
per  minute;  let  N  =  revolutions  per  minute;  let  Fr  =  torsional 
moment.  Then  F  is  the  force  factor  of  the  work  transmitted, 
and  2nrN  is  the  space  factor  of  the  work  transmitted.  Hence 

A 
2FnrN  =  A ,     and      Fr  =  torsional  moment  =  — ^ 

271 N 

If  the  work  is  to  be  transmitted  to  another  shaft  by  means  of 
a  spur-gear  whose  radius  is  r\,  then  for  equilibrium  F\r\=Fr, 

Fr 

and  FI  = — .     FI  is  the  force  at  the  pitch  surface  of  the  gear 

whose  radius  is  r-i,  i.e.,  it  is  the  force  to  be  sustained  by  the  gear- 
teeth.  Hence,  in  general,  the  force  sustained  by  the  teeth  o]  a  gear 
equals  the  torsional  moment  divided  by  the  pitch  radius  o]  the  gear. 

When  the  maximum  force  to  be  sustained  is  known  the  teeth 
may  be  given  proper  proportions.  The  dimensions  upon  which 
the  tooth  depends  for  strength  are :  Thickness  of  tooth  =  /,  width 
of  face  of  gear  =  b,  and  depth  of  space  between  teeth  =  /.  These 
all  become  known  when  the  pitch  is  known,  because  /  is  fixed  for 
any  pitch,  and  /  and  b  have  values  dictated  by  good  practice. 
The  value  of  b  may  be  varied  through  quite  a  range  to  meet  the 
requirements  of  any  special  case. 

In  computations  for  strength  the  tooth  is  treated  as  a  canti- 
lever. It  has  been  customary  to  consider  that  the  entire  load 
is  borne  by  a  single  tooth  (i.e.,  that  there  is  contact  between  only 
one  pair  of  teeth),  the  point  of  application  being  the  extreme 
end  of  the  tooth,  and  the  direction  of  the  acting  force  being 
normal  to  the  tooth  profile  at  that  point.  This  assumes  that 
the  arc  of  action  is  no  greater  than  the  pitch  arc,  which  may 
be  true  of  a  pair  of  gears  having  a  low  number  of  teeth;  but 


372  MACHINE    DESIGN. 

in  all  cases  in  which  the  arc  of  action  exceeds  the  pitch  arc  the 
force  is  borne  by  several  pairs  of  teeth  in  simultaneous  engage- 
ment, and  to  consider  it  borne  by  a  single  pair  leads  to  an  excess 
of  strength.  This  is  of  course  an  assumption  on  the  safe  side. 
Experimentally  determined  coefficients  to  correct  for  the  ratio  of 
the  arc  of  action  to  the  pitch  arc  will  be  found  in  Table  XXXVIII. 
It  is  also  assumed  that  the  load  is  uniformly  distributed  across 
the  face  of  the  tooth.  This  is  a  safe  assumption  if  the  width  of 
face,  b,  does  not  exceed  three  times  the  circular  pitch,  i.e.,  ^P,  and 
if  the  gears  are  well  aligned  and  rigidly  supported.  All  teeth  of 
the  same  pitch  have  not  the  same  form,  as  was  explained  in  the 
discussion  of  interchangeable  gears,  and  therefore  they  vary  in 
strength.  The  fewer  teeth  the  thinner  they  will  be  at  their  base 
and  consequently  the  weaker  they  will  be  when  acting  as  canti- 
levers. 

T, 


FIG.  219.  FIG.  220. 

Mr.  Wilfred  Lewis  *  has  drawn  a  number  of  figures  on  a 
large  scale  to  represent  very  accurately  the  teeth  cut  by  complete 
sets  of  cutters  of  the  15°  involute,  the  20°  involute,  and  the  cycloidal 
systems.  In  the  latter  he  used  a  rolling  circle  having  a  diameter 
equal  to  the  radius  of  the  12-tooth  pinion.  The  proportions 
of  the  teeth  used  in  his  investigation  are  slightly  different  from 
those  given  above  which  correspond  to  the  Brown  and  Sharpe 
system,  but  no  serious  errors  will  result  from  applying  the  formulas 
derived  by  him.  His  reasoning  was  as  follows  (see  Fig.  221): 

*  Proc.  Phila.  Eng.  Club,  1893,  and  Amer  Mach.,  May  4  and  June  22,  1893. 


TOOTHED  WHEELS   OR   GEARS.  373 

The  greatest  stress  in  the  tooth  occurs  when  the  load  is  applied 
at  the  end  of  the  tooth  as  indicated  by  the  arrow  at  a,  its  direction 
of  action  being  normal  to  the  profile  at  a.  The  component  of 
this  force,  which,  is  effective  to  produce  rotation  of  the  gear,  equals 
F  and  is  called  the  working  force. 

This  load  is  applied  at  b  and  induces  a  transverse  stress  in 
the  tooth.  To  determine  where  the  tooth  is  weakest  advantage 
is  taken  of  the  fact  that  any  parabola  in  the  axis  be  and  tangent  to 
bF  incloses  a  beam  of  uniform  strength.  Of  all  the  parabolas 
that  may  thus  be  drawn  one  only  will  be  tangent  to  the  tooth  form 
(as  shown  by  the  dotted  line  in  the  figure)  and  the  weakest  sec- 
tion of  the  tooth  will  be  that  through  the  points  of  tangency  c 
and  d.  Having  determined  the  weakest  section  in  each  case,  Mr. 
Lewis  developed  the  following  general  formulae  from  the  data  so 
obtained : 

For  15°  involute  and  the  cycloidal  system,  using  a  rolling 
circle  whose  diameter  equals  the  radius  of  the  i2-tooth  pinion, 

/  0.684  \ 

F  =  jPb(o.i24— -jy-1. 

For  the  20°  involute  system 


F=  working  force  in  pounds; 

/=safe  allowable  unit  stress  in  pounds  per  square  inch; 
P=  circular  pitch  in  inches; 

b  -width  of  face  of  gear  in  inches; 
N  =  number  of  teeth  in  the  gear. 

*  For  the  cycloidal  system,  using  a  rolling  circle  whose  diameter  equals  the 
radius  of  the  i5-tooth  pinion, 

o.678 


>=fPb(o. 


I06 


(Trans.  A.  S.  M.  E.,  Vol.  XVIII,  p.  776.) 


374 


MACHINE  DESIGN. 


Experimental  data  fixing  the  value  of /for  different  materials 
and  velocities  have  been  lacking. 

Experiments  made  at  Stanford  University  *  lead  to  the  fol- 
lowing equation  for  14^°  standard,  involute,  cast  iron,  cut  gears: 


F  = 


K 


I.26 

0.154 - 


JF  =  safe  working  load  at  pitch  line,  in  pounds; 
/6  =  modulus  of  rupture  in  flexure 

=  36,000  Ibs.  per  square  inch  for  cast  iron; 
P  =  circular  pitch,  inches; 
b  =  width  of  face,  inches ; 
K  =  factor  of  safety 

=  4  for  uniform  stress  in  one  direction  only 

=  6  for  suddenly  applied  loads,  one  direction  only 

=  8  for  shocks  and  reversals  of  stress ; 
v  =  velocity  coefficient  from  Table  XXXIV 
ct  =  arc  of  action  coefficient  from  Table  XXXV; 
N  =  number  of  1  eeth  in  gear. 

TABLE  XXXIV. — VELOCITY  COEFFICIENTS,  142°  INVOLUTE  GEARS. 


Pitch  velocity  in  feet 
per  minute  
Coefficient  v.    ... 

o 
i  .00 

IOO 

0.80 

200 
0.73 

300 
o.(8 

400 
0.64 

500 
0.60 

750 
o-S5 

1000 

0.50 

1250 
o-45 

1500 
0-43 

1750 
0.41 

2OOO 
0.40 

TABLE  XXXV. — ARC  OF  ACTION  COEFFICIENTS,  14^°  INVOLUTE  GEARS. 


Arc  of  action 

Pitch  arc 
Coefficient  ot    

1 

1.4 
i  .os 

1.6 

I  .  IO 

i-7 
i  .15 

1.8 
1.24 

1.9 
1.38 

i-95 
i  .47 

2.OO 
I  .60 

2.20 
T    60 

*  The  Strength   of  Gear  Teeth,   G.  H.  Marx,   Trans.  A.  S.  M.  E.,  1912;    and 
G.  H.  Marx  and  L.  E.  Cutter,  Trans.  A.S.M.E,  1915. 


TOOTHED   WHEELS  OR   GEARS. 


375 


For  20°  involute,  stub-tooth,  cast-iron,  cut  gears  the  same 
investigation  yielded  the  formula : 


The  values  of  v  and  a  to  be  used  in  this  equation  are  given  in 
Tables  XXXVI  and  XXXVII. 

TABLE  XXXVI. — VELOCITY  COEFFICIENTS,  20°  INVOLUTE,  STUB-TOOTH  GEARS. 


Pitch  vel.,  ft.  per 
min  

o 

IOO 

200 

300 

400 

500 

750 

IOOO 

1250 

1500 

1750 

2OOO 

Coefficient  v 

I  .CO 

0.83 

0.76 

0.71 

0.67 

0.64 

o-59 

0.55 

0.52 

0.50 

0.47 

0-45 

TABLE  XXXVII. — ARC  OF  ACTION  COEFFICIENTS,  20°  INVOLUTE,  STUB-TOOTH 

GEARS. 


Arc  of  action 

T      Af\ 

T    ef\ 

T      fit 

Pitch  arc 

i  .  23 

x-37 

J-43 

1  .47 

1.49 

r-53 

1  .50 

Coefficient,  a  

I    OO 

i  .  i^ 

i  .  20 

i  .  24 

I  .  2^ 

T    ?6 

i  .  27 

i  .  20 

I    31 

I      33 

To  save  computation  of  the  arc  of  action,  Table  XXXVIII 
gives  the  values  of  a  for  both  forms  for  typical  sets  of  gears. 
Interpolations  can  be  made  readily  for  other  combinations. 


TABLE  XXXVIII.— VALUES  OF  a. 


Corresponding  a. 


reetn  in  engaging  Lrears. 

i4i°    Invol. 

20°  Invol.    Stub 
Tooth. 

Single  tooth  engages 

.OO 

.00 

12 

12 

.00 

•13 

20 

30 

•15 

.20 

30 

30 

•47 

.  22 

30 

40 

.60 

.24 

3° 

60 

.60 

•25 

30 

80 

.60 

.26 

30 

IOO 

.60 

.27 

30 

Rack 

.60 

.29 

IOO 

IOO 

.60 

•31 

IOO 

Rack 

.60 

•33 

376  MACHINE    DESIGN. 

Experiments  reported  by  Mr.  Andrew  Gleason  *  show  the 
results  of  tests  on  14  tooth  steel  pinions,  14^°  involute,  i-inch 
face,  of  various  kinds  of  steel,  soft,  case-hardened,  and  tempered. 
His  results  indicate  that  for  soft,  0.20  carbon,  open-hearth 
steel  a  value  of /6  of  60,000  will  not  exceed  the  elastic  limit.  The 
same  material  case  hardened  and  heat  treated  showed  an  ultimate 
breaking  value  of  fb  in  excess  of  170,000.  Nickel  steel  gave 
values  of  /6  about  15  per  cent  higher  than  these;  and  chrome- 
nickel  tempering  steel  values  about  50  per  cent  higher.  Limited 
in  number  though  they  are,  these  experiments  are  very  significant. 

For  rawhide  gears,  -^  may  be  taken  as  5000  Ibs.  per  square 
A 

inch  as  a  minimum.     Hard  fibre  is  more  brittle;   -^  =  5000  may 

K. 

be  taken  as  a  maximum. 

In  these  formulae,  for  a  given  gear  the  whole  right  side  of 
the  equation  becomes  known  and  the  allowable  value  of  F  is 
readily  determined.  It  is  more  difficult  to  apply  the  formulae 
where  the  force  to  be  transmitted  is  given.  In  such  a  case  the 
value  of  P  is  determined  by  trial. 

210.  Problem. — Design  a  pair  of  14^°  involute  gears  to  transmit 
6  H.P.  The  distance  between  centers  is  10  inches  and  the 
velocity  ratio  of  the  shafts  is  to  be  §.  The  pinion  shaft  makes 
150  revolutions  per  minute. 

The  distance  between  centers  being  10  inches  and  the  velocity 
ration  f,  the  radii  will  be  to  each  other  as  3:2  and  their  sum 
=  10  inches,  hence  the  radius  of  the  pinion  will  be  4  inches,  while 
that  of  the  gear  will  be  6  inches.  If  both  gears  are  of  the  same 
material  the  teeth  of  the  smaller  will  be  the  weaker.  Com- 
putations will  therefore  be  made  for  the  pinion  because  the  gear 
will  be  stronger  and,  consequently,  safe.  The  pitch  diameter 

*  Machinery,  Jan.,  1914. 


TOOTHED   WHEELS    OK   GEARS.  377 


of  the  pinion  =8  inches,  its  velocity  =  150  X^X^  =314.2  feet  per 
minute. 

6  H.P.  =6X33,  ooo  =  198,000  ft.-lbs.  per  minute; 

198000 

F  =  —  -  =632  Ibs. 
314.2 

Assuming  cast   iron,    14^°   involute  gears,   variable  load   in 
one  direction  only: 


Ibs.  P  is  sought,  b  is  unknown,  but  a  trial  value, 
proportionate  to  the  size  of  the  gear  diameters,  may  be  assumed; 
if  the  value  assumed  is  too  small  it  leads  to  an  irrational  quadratic 
equation  for  P;  if  too  great  a  value  is  used  P  will  be  less  than 
%b.  In  this  case  assume  first  trial  value  of  b=i  inch.  N  is 

unknown,  so  —  is  substituted  for  it,  D\  being  the  pitch  diameter 

of  the  pinion,  8  inches  in  this  case.  The  velocity  being  314.2 
feet  per  minute,  the  value  of  the  velocity  coefficient,  v,  from  Table 
XXXIV,  is  0.68.  The  ratio  of  arc  of  action  to  pitch  arc  being 
unknown  it  is  safe  to  take  a  =  i.oo.  /6  =  36,000  Ibs.  per  square 
inch.  K  =  6. 

36,oooXPXi/  1.26  XP\ 

632  =  --  -  -  (.IS4-__J0.68Xi, 

-0.6184, 


/.   P -i. 54=  ±V -0.6184. 

This  shows  that  the  assumed  trial  value  of  b=i  inch  was 
too  ^mall.     Try  b  =  2  inches. 

36,oooXPX2/  i.26P\ 

632  =  ^-  —    .154-        -0.68X1. 

6  \  25.137 


378  MACHINE  DESIGN. 


/.   P  =  o.63  (smaller  root). 

Nearest  regular  diametral  pitch  p  =  $. 
This  gives  5  X  8  =  40  teeth  on  pinion,  and 
5  X  12  =  60  teeth  on  gear. 

These  values  will  answer,  although  the  ratio  of  b  to  P  is  a 
little  more  than  the  maximum  desirable  value  of  3.  A  new  trial 
of  b=i%  inches  would  lead  to  a  value  of  P  of  about  0.92.  The 
nearest  regular  value  of  p  would  be  3^,  giving  the  pinion  and 
gear,  28  and  42  teeth,  respectively. 

If  the  greatest  allowable  value  of  b  still  leaves  the  imaginary, 
then  the  value  of  fb  must  be  increased  either  by  using  a  stronger 
material  or*  by  cutting  down  the  factor  of  safety. 

Another  way  of  solving  the  problem  would  have  been  to  assume 
b  in  terms  of  P,  thus  b  =  cP  (c  being  less  than  3),  thus  giving 

N 


.. 

In  this,  substitute  trial  values  for  P  until  one  is  found  which 
satisfies  the  equation. 

211.  Tooth  Friction,  Pressure  and  Abrasion.*  —  From  the 
nature  of  their  relative  motion  there  is  a  sliding,  under  pressure, 
of  the  engaging  tooth  surfaces  over  each  other.  The  distance 
rubbed  over  in  any  case  while  a  pair  of  teeth  are  in  engagement 
can  be  computed  from  the  fact  that  the  relative  sliding  is  equal 
to  the  sum  of  the  addendum  of  the  driver  less  the  engaging  length 
of  the  driven  dedendum  plus  the  addendum  of  the  driven  tooth, 
less  the  engaging  length  of  the  driver  dedendum.  This  dis- 

*  For  exhaustive  treatment  see  paper  by  Lasche  in  Z.  d.  V.  d.  I.,  Vol.  XLIII, 
1899,  and  by  Buchner,  Z.  d.  V.  d.  I.,  Vol.  XL  VI,  1902.  Also  Bach's  Maschinen 
Elemente,  nth  ed.,  pp.  304-336. 


TOOTHED    WHEELS   OR    GEARS.  379 

tance  multiplied  by  /dPv,  where  /*  equals  the  coefficient  of  friction 
and  Ff  equals  the  average  normal  pressure,  gives  the  friction  work 
on  each  pair  of  teeth  during  one  engagement.  There  is  no 
difficulty  in  mathematically  determining  the  space  factor  for  any 
given  form  and  proportions  of  engaging  teeth.  A  value  of  fi 
depending  upon  the  nature  of  the  surfaces,  the  lubricant,  the 
temperature,  whether  there  is  bath  lubrication  or  not,  etc.,  may 
be  assumed  with  a  close  approximation  to  the  probable  actual 
value;  but  Ff  will  remain  a  variable  term  dependent  not  only 
upon  F,  the  tangential  force  at  the  pitch  circle  contact,  but  also 
upon  the  number  of  teeth  simultaneously  in  action,  and  the 
form  of  the  path  of  the  point  of  tooth  contact.  Experimental 
results  show  that  with  accurately  cut,  properly  mounted,  well 
lubricated,  not  overloaded  gears  this  tooth  friction  loss  is  so  little 
as  to  be  practically  negligible.  Efficiencies  of  99  per  cent  are 
not  unusual. 

The  question  of  the  allowable  tooth  pressure  to  avoid  squeezing 
out  of  the  lubricant  and  consequent  abrasion,  or  even  permanent 
elastic  deformation  without  removal  of  the  lubricant,  is  one  still 
in  need  of  experimental  investigation. 

It  is  evident  that  the  allowable  pressure  will  depend  upon  the 
properties  of  the  lubricant  and  the  method  of  its  application. 
It  is  also  evident  that  it  will  depend  upon  the  radii  of  curvature 
of  the  engaging  tooth  profiles  as  well  as  upon  the  strength,  elastic 
yielding  and  surface  hardness  of  the  tooth  materials. 

The  real  factor  governing  the  economic  selection  of  gear- 
tooth  materials  and  proportions  will  be  this  one  of  allowable 
pressure  and  relative  wear  rather  than  the  mere  ultimate  strength. 
Various  formulae  for  allowable  pressure  have  been  proposed. 
Lasche  *  gives  charts  for  raw-hide  on  cast-iron,  and  bronze  on 
steel  gears,  which  lead  to  the  following  equations: 

*  Z.  d.  V.  d.  I.,  Vol.XLIII,  pp.  1490-1491. 


38°  .         MACHINE  DESIGN. 

For  raw-hide  on  cast-iron  (cut  teeth), 
FXr.p.m. 

- 


I?     i  -...   . 

For  bronze  on  steel,  —  —  £  60,000. 

eXb 

F  =  tangential  force  at  pitch  line,  pounds; 
r.p.m.  =  revolutions  per  minute; 

e  =  number  of  pairs  of  teeth  in  simultaneous  engagement  = 
arc  of  action 

pitch  arc 
b  =  width  of  face,  inches. 

These  equations  are,  however,  only  of  value  for  comparable 
conditions  —  continuous  uniform  service  (motor  gears),  regularity 
of  speed,  good  workmanship,  accurately  cut  gears,  etc.  Bach 
proposes  the  equation  for  cut,  cast-iron  gears  for  continuous 
service 

F=Pb(2S4.$  -y.iVr.p.m.). 

P  =  circular  pitch,  inches.    F  and  b  as  above. 

This  gives  values  to  F  which  seem  too  small  for  well-executed 
installations.  It  would  make  F  =  o,  for  any  cast-iron  gears  running 
at  1600  r.p.m.  or  over,  which  is  contrary  to  experience.  The  whole 
subject  needs  further  accurate  and  wide-reaching  experimentation. 

212.  Non-circular  Wheels.  —  Only  circular  centrodes  or  pitch 
curves  correspond  to  a  constant  velocity  ratio;  and  by  making 
the  pitch  curves  of  proper  form,  almost  any  variation  in  the  velocity 
ratio  may  be  produced.  Thus  a  gear  whose  pitch  curve  is  an 
ellipse,  rotating  about  one  of  its  foci,  may  engage  with  another 
elliptical  gear,  and  if  the  driver  has  a  constant  angular  velocity 
the  follower  will  have  a  continually  varying  angular  velocity. 
If  the  follower  is  rigidly  attached  to  the  crank  of  a  slider-crank 
chain,  the  slider  will  have  a  quick  return  motion.  This  is  some- 
times used  for  shapers  and  slotting-machines.  When  more  than 


TOOTHED  WHEELS   OR    GEARS.  381 

one  fluctuation  of  velocity  per  revolution  is  required,  it  may  be 
obtained  by  means  of  "lobed  gears";  i.e.,  gears  in  which  the 
curvature  of  the  pitch  curve  is  several  times  reversed.  If  a 
describing  circle  be  rolled  on  these  non -circular  pitch  curves,  the 
tooth  outlines  will  vary  in  different  parts;  hence  in  order  to  cut 
such  gears,  many  cutters  would  be  required  for  each  gear.  Prac- 
tically this  would  be  too  expensive;  and  when  such  gears  are 
used  the  pattern  is  accurately  made,  and  the  cast  gears  are  used 
without  "tooling"  the  tooth  surfaces. 

213.  Step,  Twisted  and  Herring-bone  Gears. — If  a  pair  of 
wide-faced,  ordinary  spur  gears  be  divided  into  several  pairs 
of  narrow-faced  gears  and  then  the  successive  gears  on  the  one 
shaft  be  rotated  on  the  shaft  by  uniformly  progressive  angles, 
the  mating  gears  will  be  rotated  through  corresponding  linear 
distances  and,  therefore,  through  proportionate  angles.  Each 
pair  of  narrow  gears  will  mesh  as  before,  but  it  is  clear  that  the 
interval  of  shaft  rotation  between  successive  pairs  of  teeth  coming 
into  (and  going  out  of)  mesh  will  be  reduced  in  proportion  to 
the  number  of  narrow  or  step  gears.  This  leads  to  more  con- 
tinuous and  smooth  action. 

If  the  original  gears  be  considered  as  made  up  of  a  series 
of  very  thin  disks  or  laminae,  the  uniform  increment  of  angular 
advance  causes  each  tooth  element,  originally  parallel  to  the 
shaft  axis,  to  become  a  helix — the  steps  having  become  infinitesimal 
in  width.  This  gives  the  teeth  a  spiral  form — their  profiles  in 
a  plane  normal  to  their  axes  of  rotation  being  those,  however, 
of  the  original  spur  gears.  If  the  twist  is  in  one  direction  only 
on  each  gear  there  will  be  an  end  thrust  on  the  shafts  when 
the  gears  are  transmitting  power,  in  addition  to  the  regular 
radial  and  tangential  forces.  If  the  gears  are  made  double,  or 
herringbone,  this  end  thrust  balances  itself.  Such  gears  are 
now  very  accurately  cut  or  hobbed  and  are  applicable  to  cases 
involving  large  velocity  ratios,  high  speeds,  and  fairly  large 


382  MACHINE  DESIGN. 

amounts  of  power.  It  is  claimed  for  them  that  they  are  free 
from  backlash,  vibration  and  objectionable  noise,  and  very  high 
efficiencies  (up  to  99  per  cent)  are  guaranteed.  These  gears 
are  for  connecting  parallel  axes  and  must  not  be  confused  with 
spiral  gears  for  non-parallel  axes.  For  these  see  the  section 
"  Spiral  Gearing." 

214.  Bevel-gears. — All  transverse  sections  of  spur-gears  are 
the  same,  and  their  axes  intersect  at  infinity.  Spur-gears  serve 
to  transmit  motion  between  parallel  shafts.  It  is  necessary  also 
to  transmit  motion  between  shafts  whose  axes  intersect.  In  this 
case  the  pitch  cylinders  become  pitch  cones;  the  teeth  are  formed 

upon  these  conical  surfaces,  the  result- 
ing gears  being  called  bevel-gears. 
To  illustrate,  let  a  and  b,  Fig.  222,  be 
the  axes  between  which  the  motion  is 
to  be  transmitted  with  a  given  velocity 
v  ratio.  This  ratio  is  equal  to  the  ratio 

rlG.  222. 

of  the  length  of  the  line  A  to  that  of  B. 

Draw  a  line  CD  parallel  to  a,  at  a  distance  from  it  equal  to 
the  length  of  the  line  A.  Also  draw  the  line  CE  parallel  to  b, 
at  a  distance  from  it  equal  to  the  length  of  the  line  B.  Join 
the  point  of  intersection  of  these  lines  to  the  point  O,  the 
intersection  of  the  given  axes.  This  locates  the  line  OF,  which 
Is  the  line  of  contact  of  two  pitch  cones  which  will  roll  together 

to  transmit  the  required  velocity  ratio.     For  j^,  =  — ,  and  if  it 

be  supposed  that  there  are  frusta  of  cones  so  thin  that  they  may 
be  considered  cylinders,  their  radii  being  equal  to  MC  and  NC, 
it  follows  that  they  would  roll  together,  if  slipping  be  prevented, 
to  transmit  the  required  velocity  ratio.  But  all  pairs  of  radii 

71  /f  /"* 

of  these  pitch  cones  have  the  same  ratio,  ^TrT^  and  therefore 
any  pair  of  frusta  of  the  pitch  cones  may  be  used  to  roll  together 


TOOTHED    WHEELS   OR   GEARS.  383 

for  the  transmission  of  the  required  velocity  ratio.  To  insure 
this  result,  slipping  must  be  prevented,  and  hence  teeth  are 
formed  upon  the  selected  frusta  of  the  pitch  cones.  The  theo- 
retical determination  of  these  may  be  explained  as  follows: 

215.  ist.  Cycloidal  Teeth. — If  a  cone,  A  (Fig.  223),  be  rolled 
upon  another  cone,  B,  their  apexes  coinciding,  an  element  be  of 
the  cone  A  will  generate  a  conical  surface,  and  a  spherical  sec- 


tion of  this  surface,  adb,  is  called  a  spherical  epicycloid.  Also 
if  a  cone,  A  (Fig.  224),  roll  on  the  inside  of  another  cone,  C, 
their  apexes  coinciding,  an  element  be  of  A  will  generate  a  conical 
surface,  a  spherical  section  of  which,  bda,  is  called  a  spherical 
hypocycloid.  If  now  the  three  cones,  B,  C,  and  A,  with  apexes 
coinciding,  roll  together,  always  tangent  to  each  other  on  one 
line,  as  the  cylinders  were  in  the  case  of -spur-gears,  there  will 
be  two  conical  surfaces  generated  by  an  element  of  A :  one  upon 
the  cone  B  and  another  upon  the  cone  C.  These  may  be  used 
for  tooth  surfaces  to  transmit  the  required  constant  velocity 
ratio.  Because,  since  the  line  of  contact  of  the  cones  is  the  axo  * 
of  the  relative  motion  of  the  cones,  it  follows  that  a  plane  normal 
to  the  motion  of  the  describing  element  of  the  generating  cone 
at  any  time  will  pass  through  this  axo.  And  also,  since  the 
describing  element  is  always  the  line  of  contact  between  the  gen- 
erated tooth  surfaces,  the  normal  plane  to  the  line  of  contact  of 
the  tooth  surfaces  always  passes  through  the  axo,  and  the  condi- 
tion of  rotation  with  a  constant  velocity  ratio  is  fulfilled. 

216.  2d.  Involute  Teeth. — If  two  pitch  cones  are  in  contact 
along  an  element,  a  plane  may  be  passed  through  this  element 

*  An  axo  is  an  instantaneous  axis,  of  which  a  centre  is  a  projection. 


MACHINE  DESIGN. 


making  an  angle  (say  75°)  with  the  plane  of  the  axes  of  the  cones. 
Tangent  to  this  plane  there  may  be  two  cones  whose  axes  coin- 
cide with  the  axes  of  the  pitch  cones.  If  a  plane  is  supposed  to 
wind  off  from  one  base  cone  upon  the  other,  the  line  of  tangency 
of  the  plane  with  one  cone  will  leave  the  cone  and  advance  in  the 
plane  toward  the  other  cone,  and  will  generate  simultaneously 
upon  the  pitch  cones  conical  surfaces,  and  spherical  sections  of 
these  surfaces  will  be  spherical  involutes.  These  surfaces  may 
be  used  for  tooth  surfaces,  and  will  transmit  the  required  con- 
stant velocity  ratio,  because  the  tangent  plane  is  the  constant 
normal  to  the  tooth  surfaces  at  their  line  of  contact,  and  this 
plane  passes  through  the  axo  of  the  pitch  cones. 

217.  Determination  of  Bevel-gear  Teeth. — To  determine  the 
tooth  surfaces  with  perfect  accuracy,  it  would  be  necessary  to 
draw  the  required  curves  on  a  spherical  surface,  and  then  to 
join  all  points  of  these  curves  to  the  point  of  intersection  of  the 
axes  of  the  pitch  cones.  Practically  this  would  be  impossible, 
and  an  approximate  method  is  used. 

If  the  frusta  of  pitch  cones  be  given,  B  and  C,  Fig.  225,  then 
points  in  the  base  circles  of  the  cones,  as  L,  M,  and  K,  will  move 
always  in  the  surface  of  a  sphere  whose  projection  is  the  circle 


A 


FIG.  225. 


FIG.  226. 


LA  KM.  Properly,  the  tooth  curves  should  be  laid  out  on  the 
surface  of  this  sphere  and  joined  to  the  center  of  the  sphere  to 
generate  the  tooth  surfaces.  Draw  cones  LGM  and  MHK  tan- 


TOOTHED   WHEELS   OR  GEARS.  385 

gent  to  the  sphere  on  circles  represented  in  projection  by  lines 
LM  and  MK.  They  are  called  the  "back  cones."  If  now 
tooth  curves  are  drawn  on  these  cones,  with  the  base  circle 
of  the  cones  as  pitch  circles,  they  will  approximate  the  tooth 
curves  that  should  be  drawn  on  the  spherical  surface.  But  a 
cone  may  be  cut  along  one  of  its  elements  and  rolled  out,  or  de- 
veloped, upon  a  plane.  Let  MDH  be  a  part  of  the  cone  MHK, 
developed,  and  let  MNG  be  a  part  of  the  cone  MGL,  developed. 
The  circular  arcs  MD  and  M N  may  be  used  just  as  pitch  circles 
are  in  the  case  of  spur-gears,  and  the  teeth  may  be  laid  out  in 
exactly  the  same  way,  the  curves  being  either  cycloidal  or  in- 
volute, as  required.  Then  the  developed  cones  may  be  wrapped 
back  and  the  curves  drawn  may  serve  as  directrices  for  the 
tooth  surfaces,  all  of  whose  elements  converge  to  the  center  of 
the  sphere  of  motion. 

218^  Cutting  Bevel-gear  Teeth. — The  teeth  of  spur-gears  may 
be  cut  by  means  of  milling-cutters,  because  all  transverse  sections 
are  alike,  but  with  bevel-gears  the  conditions  are  different.  The 
tooth  surfaces  are  conical  surfaces,  and  therefore  the  curvature 
varies  constantly  from  one  end  of  the  tooth  to  the  other.  Also  the 
thickness  of  the  tooth  and  the  width  of  space  vary  constantly 
from  one  end  to  the  other.  But  the  curvature  and  thickness 
of  a  milling-cutter  cannot  vary,  and  therefore  a  milling-cutter 
cannot  cut  an  accurate  bevel-gear.  Small  bevel-gears  are, 
however,  cut  with  milling-cutters  with  sufficient  accuracy  for 
practical  purposes.  The  cutter  is  made  as  thick  as  the  narrowest 
part  of  the  space  between  the  teeth,  and  its  curvature  is  made 
that  of  the  middle  of  the  tooth.  Two  cuts  are  made  for  each 
space.  Let  Fig.  226  represent  a  section  of  the  cutter.  For  the 
first  cut  it  is  set  relatively  to  the  gear  blank,  so  that  the  pitch 
point  a  of  the  cutter  travels  toward  the  apex  of  the  pitch  cone, 
and  for  the  second  cut  so  that  the  pitch  point  b  travels  toward 
the  apex  of  the  pitch  cone.  This  method  gives  an  approximation 


386  MACHINE  DESIGN. 

to  the  required  form.  Gears  cut  in  this  manner  usually  need 
to  be  filed  slightly  before  they  work  satisfactorily.  Bevel-gears 
with  absolutely  correct  tooth  surfaces  may  be  made  by  planing. 
Suppose  a  planer  in  which  the  tool  point  travels  always  in  some 
line  through  the  apex  of  the  pitch  cone.  Then  suppose  that  as  it 
is  slowly  fed  down  the  tooth  surface,  it  is  guided  along  the  required 
tooth  curve  by  means  of  a  templet.  From  what  has  preceded 
it  will  be  clear  that  the  tooth  so  formed  will  be  correct.  Planers 
embodying  these  principles  have  been  designed  and  constructed 
by  Mr.  Corliss  of  Providence,  and  Mr.  Gleason  of  Rochester, 
with  the  most  satisfactory  results. 

219.  Design  of  Bevel-gears. — Given  energy  to  be  transmitted, 
rate  of  rotation  of  one  shaft,  velocity  ratio,  and  angle  between 
axes;  to  design  a  pair  of  bevel-gears.  Locate  the  intersection 
of  axes,  O,  Fig.  227.  Draw  the  axes  OA  and  OB,  making  the 
required  angle  with  each  other.  Locate  OC,  the  line  of  tangency 
of  the  pitch  cones,  by  the  method  given  on  p.  382.  Any  pair  of 
frusta  of  the  pitch  cones  may  be  selected  upon  which  to  form 
the  teeth.  Special  conditions  of  the  problem  usually  dictate  this 
selection  approximately.  Suppose  that  the  inner  limit  of  the 
teeth  may  be  conveniently  at  D.  Then  make  DP,  the  width 
of  face,  =DO+2.  Or,  if  P  is  located  by  some  limiting  condi- 
tion, lay  off  PD=PO--$.  In  either  case  the  limits  of  the  teeth 
are  denned  tentatively.  Now  from  the  energy  and  the  number 
of  revolutions  of  one  shaft  (either  shaft  may  be  used)  the  moment 
of  torsion  may  be  found.  The  mean  force  at  the  pitch  surface 
=  this  torsional  moment -^ the  mean  radius  of  the  gear;  i.e., 
the  radius  of  the  point  M,  Fig.  227,  midway  between  P  and  D. 
The  pitch  corresponding  to  this  force  may  be  found. 

In  order  to  compute  it  consider  the  teeth  of  the  pinion  (i.e., 
the  smaller  gear),  as  they  will  be  the  weaker.  Having  found 
the  force,  F,  which  is  to  be  transmitted  we  determine  the  pitch 
required  to  carry  F  by  a  spur-gear,  whose  pitch  radius  =  MN 


TOOTHED    WHEELS  OR   GEARS. 


387 


and  whose  width  of  face,  b,  equals  PD.  (The  radius  MN  is 
used  to  govern  the  shape  of  the  tooth  and  not  MR,  because  the 
teeth  are  laid  off  on  the  developed  back  cones  and  not  on  the 
pitch  circles,  as  has  been  explained.  The  circle  whose  radius 
is  MN  is  sometimes  called  the  formative  circle  in  order  to  dis- 
tinguish it  from  the  pitch  circle.) 

The  pitch  of  such  a  spur-gear  would  be  the  mean  pitch  of  the 
bevel-gears.      But  the  pitch  of  bevel-gears  is  measured  at  the 


FIG.  227. 

large  end,  and  diametral  pitch  varies  inversely  as  the  distance 
from  O.  In  this  case  the  distances  of  M  and  P  from  O  are  to 
each  other  as  5  is  to  6.  Hence  the  value  of  diametral  pitch  found 
X|=the  diametral  pitch  of  the  bevel-gear.  If  this  value  does 
not  correspond  with  any  of  the  usual  values  of  diametral  pitch, 
the  next  smaller  value  may  be  used.  This  would  result  in  a 
slightly  increased  factor  of  safety.  If  the  diametral  pitch  thus 
found,  multiplied  by  the  diameter  2PQ,  does  not  give  an  integer 
for  the  number  of  teeth,  the  point  P  may  be  moved  outward 


388  MACHINE  DESIGN. 

along  the  line  OC,  until  the  number  of  teeth  becomes  an  integer. 
This  also  would  result  in  slight  increase  of  the  factor  of  safety. 
The  point  P  is  thus  finally  located,  the  corrected  width  of  face  = 
PO  -1-3,  and  the  pitch  is  known.  The  drawing  of  the  gears  may  be 
completed  as  follows:  Draw  AB  perpendicular  to  PO.  With  A 
and  B  as  centers,  draw  the  arcs  PE  and  PF.  Use  these  as  pitch 
arcs,  and  draw  the  outlines  of  two  or  three  teeth  upon  them,  with 
cycloidal  or  involute  curves  as  required.  These  will  serve  to  show 
the  form  of  tooth  outlines.  From  P  each  way  along  the  line  AB 
lay  off  the  addendum  and  the  clearance.  From  the  four  points 
thus  located  draw  lines  toward  O  terminating  in  the  line  DG. 
The  tops  of  teeth  and  the  bottoms  of  spaces  are  thus  defined. 
Lay  off  upon  AB  below  the  bottoms  of  the  spaces  a  space  HJ 
about  equal  to  |-  the  thickness  of  the  tooth  on  the  pitch  circle. 
This  gives  a  ring  of  metal  to  support  the  teeth.  From  /  draw 
JK  toward  O.  The  web  L  should  have  about  the  same  thickness 
as  the  ring  has  at  K.  Join  this  web  to  a  properly  proportioned 
hub  as  shown.  The  plan  and  elevation  of  each  gear  may  now 
be  drawn  by  the  ordinary  methods  of  projection.  Use  large 
fillets. 

220.  Twisted  Bevel  Gears. — In  a  manner  entirely  analogous 
to  that  explained  in  Sec.  213  concerning  the  development  of 
twisted  and  herringbone  from  ordinary  spur  gears,  a  pair  of 
bevel  gears  may  be  considered  as  made  up  of  very  thin  en- 
gaging disks  or  laminae.  These  may  be  given  progressive 
angles  of  twist,  causing  the  elements  of  the  teeth,  originally 
straight  lines  converging  at  the  center  of  relative  rotation,  to 
become  spirals  on  conical  surfaces.  Sections  on  planes  nor- 
mal to  the  axes  of  rotation  remain  of  the  same  profile  as 
ordinary  straight-toothed  bevel  gears.  The  same  claims  of 
improved  smoothness  of  action,  noiselessness,  high  efficiency 
and  strength  are  made  for  twisted  bevel  gears  as  for  twisted 
spur  gears. 


TOOTHED  WHEELS    OR    GEARS. 


389 


221.  Skew  Bevel-gears. — When  axes  which  are  parallel  are 
to  be  connected  by  gear-wheels  the  basic  form  of  the  wheels  is  the 
cylinder.  When  intersecting  axes  are  to  be  so  connected  the  basic 
form  is  the  cone  or  cone  frustum.  It  is  sometimes  necessary  to 
communicate  motion  between  axes  that  are  neither  parallel  nor 
intersecting.  If  the  parallel  axes  are  turned  out  of  parallelism, 
or  if  intersecting  axes  are  moved  into  different  planes,  so  that  they 
no  longer  intersect,  the  pitch  surfaces  become  hyperboloids  of  revo- 
lution in  contact  with  each  other  along  a  straight  line,  which  is  the 
generatrix  of  the  pitch  surfaces.  These  hyperboloids  of  revolu- 


FIG.  227^4. 

tion  rotated  simultaneously  about  their  respective  axes,  circum- 
ferential slippage  at  their  line  of  contact  being  prevented,  will 
transmit  motion  with  a  constant  velocity  ratio.  There  is,  however, 
necessarily  a  slippage  of  the  elements  of  the  surfaces  upon  each 
other  parallel  to  themselves.  Teeth  may  be  formed  on  these  pitch 
surfaces,  and  they  may  be  used  for  the  transmission  of  motion 
between  shafts  that  are  not  parallel  nor  in  the  same  plane.  Fig. 
227^4  shows,  in  plan  view,  a  pair  of  such  hyperboloids  of  revolu- 
tion. Disk  portions  of  these,  cut  anywhere  except  at  the  gorge, 
are  approximately  conical  frusta  and  are  the  basic  form  of  skew 
bevel-gears.  The  difficulties  of  construction  and  the  additional 
friction  due  to  slippage  along  the  elements  make  them  undesir- 
able in  practice,  and  there  is  seldom  a  place  where  they  cannot  be 


39o  MACHINE   DESIGN. 

replaced  by  some  other  form  of  connection.  It  is  evident  from 
the  figure,  for  instance,  that  disk  portions  taken  at  the  gorge  of 
the  hyperboloids  of  revolution  are  approximately  cylinders,  which 
are  the  basic  forms  of  ordinary  spiral  gears. 

A  very  complete  discussion  of  skew  bevel-gears  may  be  found 
in  Reuleaux's  "Constructor." 

222.  Spiral  Gearing.*  —  If  line  contact  is  not  essential 
there  is  much  wider  range  of  choice  of  gears  to  connect  shafts 
which  are  neither  parallel  nor  intersecting.  A  and  5,  Fig.  228, 
are  axes  of  rotation  in  different  planes,  both 
B  planes  being  parallel  to  the  paper.  Let  EF  and 
-A  GH  be  cylinders  on  these  axes,  tangent  to  each 
other  at  the  point  S.  Any  line  may  now  be 
drawn,  in  the  plane  which  is  tangent  to  the 
two  cylinders,  through  S  either  between  A  and 
B  or  coinciding  with  either  of  them.  This  line,  say  DS, 
may  be  taken  as  the  common  tangent  to  helical  or  screw 
lines  drawn  on  the  cylinders  EF  and  GH]  or  helical  surfaces 
may  be  formed  on  both  cylinders,  DS  being  their  common  tan- 
gent at  S.  Spiral  gears  are  thus  produced.  Each  one  is  a  por- 
tion of  a  many  threaded  screw.  The  contact  in  these  gears  is 
point  contact;  in  practice  the  point  of  contact  becomes  a  very 
limited  area. 

For  the  sake  of  simplicity  the  special  case  of  spiral  gears  with 
axes  at  90°  will  be  considered  first.  The  term  helix  angle  is  here 
taken  (as  in  the  treatment  of  all  other  screws)  as  meaning  the 
angle  of  the  mean  helix  with  a  plane  which  is  perpendicular  to 
the  axis  of  rotation.  Throughout  the  discussion  the  subscript  i 
is  used  in  reference  to  the  driver  and  the  subscript  2  in  reference 
to  the  follower. 

*  See  further  "Worm  and  Spiral  Gearing,"  by  F.  A.  Halsey.  Van  Nostrand 
Science  Series.  Also  "Worm  Gearing"  by  H.  K.  Thomas,  McGraw-Hill  Book 
Company. 


TOOTHED   WHEELS  OR   GEARS. 


391 


Let  Fig.  22&4  represent  the  plan  view  of  a  pair  of  spiral  gears 
with  axes  at  90°.  In  this  special  case  it  will  be  noted  that  the 
axis  of  the  follower  lies  in  the  plane  perpendicular  to  the  axis  of 
the  driver  and,  therefore,  that  the  helix  angle  of  the  driver  is  the 
angle  ABC,  made  by  the  teeth  of  the  driver  with  the  axis  of  the 
follower.  Call  this  helix  angle  of  the  driver  a\. 

Similarly,  the  helix  angle  of  the  follower  equals  the  angle  be- 
tween the  teeth  of  the  follower  and  the  axis  of  the  driver.  Call 
the  helix  angle  of  the  follower  a2. 


FIG.  228-8. 


Let  Fig.  228^  represent  the  development  of  both  pitch  cyl- 
inders in  the  tangent  plane.  It  will  be  noted  that  the  same  line 
is  normal  to  the  teeth  of  each  gear  and  that  their  pitches  measured 
on  the  normal  must  be  equal.  The  distance  occupied  by  a  tooth 
and  space  on  this  normal  is  called  the  normal  pitch. 

If  now  the  driver  (i)  be  moved  in  the  direction  AB,  the 
follower  (2)  will  be  forced  to  move  in  the  direction  CD.  For 
a  movement  of  the  driver  equal  to  ab,  the  follower  must  move  a 
distance  cb.  This  establishes  the  fact  that 

circumferential  vel.  of  follower    distance  moved  by  follower    cb 
circumferential  vel.  of  driver        distance  moved  by  driver      ab' 


cb 

But    — 
ab 


circumferential  vel.  of  follower 
circumferential  vel.  of  driver 


(i) 


392  MACHINE  DESIGN. 

Let  C= distance  between  centers  of  gears; 

DI  =  pitch  diameter  of  driver; 

DZ= pitch  diameter  of  follower; 

NI  =  number  of  teeth  of  driver; 

N%= number  of  teeth  of  follower; 
r.p.m.i  =  revolutions  per  minute  of  driver; 
r.p.m. 2  =  revolutions  per  minute  of  follower. 

The  following  equations  may  then  be  written: 


circumferential  velocity  of  follower     nDz  r.p.m.2 

—  —  = -.         .     (2) 

circumferential  velocity  of  driver       nD\  r.p.m.i 

Combining  (i)  and  (2), 

7tD2  r.p.m.2 


=  tan 


r.p.m.i 

=  r.p.m.i 
r.p.m.2 


(3) 


[NOTE. — Equation  (3)  may  also  be  written  -     —  =  — -^  tan  a\. 

r.p.m.i     DZ 

r.p.m.2  .  .          .         T          . 

But  -         -  is  the  angular  velocity  ratio.     Hence  in  spiral  gears 
r.p.m.i 

the  angular  velocity  ratio  depends  upon  two  factors:  first,  the 
inverse  of  the  ratio  of  pitch  diameters;  second,  the  tangent  of 
the  helix  angle  of  the  driver. 

In  ordinary  spur-gears,  it  will  be  remembered,  the  angular 
velocities  are  inversely  as  the  pitch  diameters,  or  the  pitch 
diameters  are  inversely  as  the  velocities.  In  spiral  gears  (axes 
at  90°)  this  is  only  true  when  tan  «i  =  i,  hence  when  the  helix 
angle  is  45°. 

Whereas  with  spur-gears,  for  a  given  distance  between  centers 
and  a  given  velocity  ratio,  the  pitch  circles  are  at  once  deter- 
mined since  there  can  only  be  a  single  pair  to  fit  the  conditions; 


TOOTHED   WHEELS   OR   GEARS. 


393 


with  spiral  gears  an  indefinite  number  of  values  may  be  used 
for  D\  and  D^.  It  is  only  necessary  to  keep  -  -  —the  given 

center  distance,  and  --tan  ai  =  the  required  velocity  ratio.     As 
D2 

DI  and  D%  are  varied  it  is  only  necessary  to  vary  tan  a\  accord- 

ingly;   or,  if  tan  a\  is   varied   -   -  must  be  varied  accordingly. 

D% 

For  gears  with  axes  at  90°,  if  0:1  =  45°,  ^2  =  45°  also,  and  the 
diameters  will  be  inversely  as  the  angular  velocities,  i.e.,  in- 
versely as  the  revolutions  per  minute.  For  all  other  values  of  a\ 
this  does  not  hold.] 

Since  C=  distance  between  centers 


2C',     .......     (4) 

/.  D2  =  2C-Di.     .......     (5) 

Substituting  in  (3), 

2C—  DI     r.p.m.i 


r.p.m.2 


tan 


r.p.m.2 


,          /      r.p.m.i  \ 

=  L>i(  i  +  -        -  tan  «i  I, 
\       r.p.m.2  / 

(6) 


.p.m.2 
2C 


r.p.m.i 
i  +  -       -tanai 
r.p.m.2 

Equations  (6)  and  (5)  will  give  us  all  the  possible  solutions  for 

r.p.m.i 

any  given  values  of  C  and  . 

r.p.m.2 


394 


MACHINE  DESIGN. 


The  particular  solution  which  will  best  suit  a  given  case  is 
determined  by  practical  considerations. 

First,  it  must  be  remembered  that  spiral  gears  have  screw 
action  and  hence  have  highest  efficiencies  for  helix  angles  whose 
value  is  in  the  neighborhood  of  45°.  For  satisfactory  operation 
it  must  never  be  allowed  to  be  less  than  15°  or  over  75°. 

Second,  it  must  be  remembered  that  as  «i  approaches  45° 
the  ratio  of  the  diameters  becomes  the  inverse  of  the  velocity 


Normal  Ilc'ix 


FIG.  228C. 

ratio.  For  a  very  great  velocity  ratio  this  may  make  one  of 
the  gears  too  small  and  the  other  too  large  for  practical  con- 
struction and  operation.  . 

Third,  the  gears  must  be  such  as  can  be  cut  with  stock 
cutters  in  any  universal  milling  machines.  For  this  reason 
their  normal  pitch  must  have  some  standard  value  and  the  lead 
of  the  tooth  helix  must  also  be  a  value  which  can  be  attained 
with  the  milling  machine. 

The  following  discussion  will  deal  with  this  third  condition. 
See  Fig.  228C,  A,  which  shows  a  spiral  gear  extended  axially 
to  a  length  sufficient  to  include  a  complete  tooth-helix. 


TOOTHED    WHEELS  OR   GE4RS.  395 

The  normal  helix  is  a  helix  at  right  angles  to  the  tooth  helix. 
Its  entire  length  is  that  length  which  will  include  (i.e.,  cut  across) 
all  the  teeth  of  the  gear  exactly  once. 

Let  Zi  =  length  of  normal  helix  of  driver; 
L2  =  length  of  normal  helix  of  follower; 
P  =  normal  pitch  of  each. 

From  Fig.  228C,  B,  which  shows  a  developed  pair  of  gears, 
it  can  be  seen  that 

Zi=7rZ>isinai;         (7) 

L2  =  7iD2  cosai;        (8) 


.-.  -1^—itanai (9) 

L>2       JL/2 


From  equation  (3), 

r.p.m.2 


=  —  -tanai, 
D2 


..  . 

L2    r.p.m.i 

That   is,   the  normal  helices   are   inversely   as   the   number   of 
revolutions  per  minute.     Since 

LI  =  length  of  normal  helix  of  driver, 

P  =  normal  pitch, 
and     NI  =  number  of  teeth  of  driver, 


Similarly,  L2=PN2, 


L2     N2    r.p.m.i* 

That  is,  the  numbers  of  teeth  are  inversely  as  the  numbers  of 
revolutions.     (Just  as  in  spur-gears.) 


396  MACHINE  DESIGN. 

From  (n),  N^-^^N^ 

r.p.m.x 

NI  must  be  a  whole  number,  as  must  also  AM  =  -^— — ~^i), 

\     r.p.m.2      / 
for   a   fractional   tooth   is   impossible.     Hence    a   normal   pitch 

must  be  selected  which  will  divide  both  LI  and  L2  perfectly. 

Also  this  normal  pitch  must  be  a  stock  value.     Let  p  be 
the  diametral  pitch  corresponding  to  the  normal  pitch  P.     Then 

Pt-,;  .-.  P-!. 
£-*••-?' 

£.-*  •••  ' 

and  />  must  correspond  to  some  standard  diametral  pitch. 

The  values  of  LI   and  L2  as  determined  by  equations  (7) 

and  (8)  may  not  be  such  as  will  give  tabular  values  for  p.  It 

then  becomes  necessary  to  take  the  nearest  standard  value  for 

p  to  that  obtained  from  p  =  —    '•    and  to    substitute  it  in  this 

Li 

equation,  thus  deriving  a  corrected  value  for  L\.  Similarly  for 
L2.  This  is  based  upon  the  assumption  that  NI  and  N2  are 
given.  If  p  is  given,  NI  and  N2  must  be  computed,  the  nearest 
corresponding  whole  numbers  selected,  and  LI  and  L2  corrected 
accordingly.  Interference  must  be  guarded  against. 

But   changing  LI   and   L2  involves   also   changing   DI,   D2, 
OL\t  and  a  2  as  can  be  seen  by  reference  to  (7)  and  (8). 

Li  =  xDi  sin  a'i,         (7) 

L2  =  xD2cos  ai,    ...     o     ...     (8) 
(  =  7tD2  sin  0:2). 


TOOTHED   WHEELS   OR   GEARS.  397 

If  a  i  be  not  altered,  it  is  evident  that  DI  and  D2  will  be 
altered  proportionately  with  LI  and  L2  and  this  will  change 

the  value  of  the  center  distance  C,  since  C= . 

2 

If  the  center  distance  may  be  altered  to  this  new  value  the 
solution  is  complete.  The  size  of  the  gear  blank  of  the  driver 

2 

would  be   DI-\ — ;  helix  angle,  a\\  number  of  teeth,  N\  ;  normal 

t 
(diametral)  pitch,   p.     For  the  follower  the  values  would  be  D2  + 

-;  a2(  =  9°°-«i);  NZ>  and  P- 
P 

In  most  cases,  however,  it  will  be  impossible  to  change  the 
value  of  C,  and  the  values  of  D\,  D2,  «i,  and  a2  must  all  be 
changed  to  keep  the  corrected  values  of  LI  and  L2. 

From  (7)  and  (8), 


ft— •=•- ,    Ds—      -; 

7T  Sin  «i  7T  COS  «l 

also,  Di  +  D2  =  2C .     .     (4) 


Divide  by 

TT  sm  ai 


n  cos  a\ 
Li 


L2    •  iC 

— tan  ai  =  —-  sm  «i (12) 

LI  L\ 


Using  corrected  values  of  L\  and  L2,  try  different  values  for 
a  i  until  we  get  an  identity.     This  is  the  correct  value  of  «i. 
Use  this  value  of  a\  and  the  correct  values  of  L\  and  L2  in 

LI                               L2 
D\  =  —    and     D2=—    , 

TT  sm  a\  TT  cos  a\ 

and  solve  for  corrected  values  of  D\  and  D2. 


398 


MACHINE  DESIGN. 


These  corrected  values  of  D\,  D2,  a\t  and  a2  (=90°— a i), 
together  with  N\,  N2,  and  p,  already  obtained,  fully  determine 
the  gears. 

There  remain  two  points  of  practical  importance  to  be  deter- 
mined: first,  the  "  pitch  of  the  tooth  helix;"  second,  the  particular 
cutter  of  the  determined  pitch  which  should  be  used. 

i.  By  pitch  of  the  tooth  helix  is  meant  the  axial  length 
corresponding  to  one  complete  turn  of  the  tooth  helix  about 
the  pitch  cylinder.  In  ordinary  screws  this  is  termed  the  "  lead," 
and  as  "  pitch  "  is  used  for  so  many  different  purposes  we  will 
use  the  term  "  lead." 

Referring  to  Fig.  2286*,  5,  it  is  clear  that 

leadi 

— =tan  «i, 


=  7rZ>i  tan 


=  tan  a2  = 


xD2  cot  a\. 

From  these  leads  the  gear  settings  of  the  milling  machine 
are  determined.  (See  Halsey's  "  Worm  and  Spiral  Gears " 
for  table  of  Brown  and  Sharpe  settings.) 

2.  In  ordinary  spur-gears  the  cutter  to  be  used  for  any  gear 

is  directly  determined  by  the 
number  of  teeth  of  the  gear. 
This  is  not  the  case  with  spiral 
gears. 

The  method  for  determining 

FlG-228Z}-  the  cutter    is    based    upon    the 

following  reasoning,  due  to  Prof.  Le  Conte. 

Reference  is  to  Fig.  22&D.    a  =  helix  angle  as  before. 

The  figure  shows  the  material  of  the  pitch  cylinder  extended 


TOOTHED   WHEELS   OR   GE4RS.  399 

either  side  of  the  gear;  abc  is  the  tooth  helix;  np  represents  a 
plane  normal  to  the  tooth  helix  at  b.  This  normal  plane  will 
cut  an  ellipse  from  the  pitch  cylinder.  The  minor  axis  will 
=  D,  the  diameter  of  the  pitch  cylinder.  The  major  axis  is 

determined    by    the    relation    —  —  =  sina;    /.  major    axis 

major  axis 

D 


sin  a 

If  we  cut  a  spiral  gear  in  the  same  way  as  we  cut  this  pitch 
cylinder,  selecting  the  point  b  midway  between  two  teeth,  the 
form  of  the  space  on  the  normal  plane  will  be  the  true  normal 
shape.  It  will  therefore  be  the  true  shape  of  the  cutter  to  be 
used. 

Now  the  curvature  of  the  normal  section  of  the  gear  at  the 
point  indicated  is,  of  course,  the  curvature  of  the  ellipse  at  the 
extremity  of  the  minor  axis.  And  the  cutter  to  be  used  would 
be  the  cutter  for  a  pitch  circle  having  this  curvature.  Such  a 
circle  (i.e.,  one  whose  radius  equals  the  radius  of  curvature  of 
the  ellipse  at  the  extremity  of  its  minor  axis)  is  called  an  "  oscu- 
lating circle." 

Let  p  =  radius  of  osculating  circle; 

a  =  half  of  major  axis  of  ellipse  = 


2  sin  a 


=  half  of  minor  axis  of  ellipse  =  — . 

2 

D2 


P=b= 


2 


4  sin2  a;          D 


Let  N0  =  number  of  teeth  of  normal  pitch  P  on  the  osculat- 
ing circle, 

......     (14) 


400  MACHINE  DESIGN. 

Combining  (i)  and  (2), 

(15) 


Let  N  =  actual  number  of  teeth  on  spiral  gear  of  diameter  D, 

Pf  =  actual  circular  pitch  of  spiral  gear. 
Then, 


P 

It  will  also  be  seen  that  —  ,  =  sin  a, 


p  AT    *D  sin  «  NP 

.'.  P'  =  - ,     /.  N=-         -,     /.  xD  =  -  -.      .    (17) 

sin  a  P  sin  a 


Substituting  value  of  D  of  (17)  in  (15), 

NP         N 

NQ=    .  p^-T-jr- (l8) 

sin3  aP     sin3  a 
For  the  driver,  then, 

For  the  follower, 

N0  =  ^~.       .       .  (20) 

sm3a2 

Equations  (19)  and  (20)  give  the  number  of  teeth  whose 
corresponding  cutters  should  be  used. 

This  completes  the  solution  for  spiral  gears  with  axes  at  90°. 

The  following  problem  gives  the  full  application  of  the  fore- 
going method.  The  computation  of  spiral  gears  which  will  run 
together  properly  calls  for  strictly  accurate  numerical  work  and 
the  use  of  logarithmic  tables  is  recommended. 


TOOTHED   WHEELS  OR   GEARS.  401 

223.  Problem. — Design  a  pair  of  spiral  gears  for  the  follow- 
ing conditions: 

Axes  at  90°.     C=3.375". 
r.p.m.2     i     revolutions  per  minute  of  follower 
r.p.m.i     2       revolutions  per  minute  of  driver  ' 
D\  =  pitch  diameter  of  driver; 
Z>2  =  pitch  diameter  of  follower; 
<*i  =  helix  angle  of  driver; 
«2  =  helix  angle  of  follower  (=90°— ai); 
NI  =  number  of  teeth  of  driver  =  10; 
^2  =  number  of  teeth  of  follower  =  20; 
LI  =  normal  helix  length  of  driver; 
1,2  =  normal  helix  length  of  follower. 

It  is  further  assumed  that,  for  reasons  of  construction,  it  is 
desired  to  have  the  two  gears  as  nearly  equal  in  size  as  possible. 

First  Solution. 

Let  Di  =  D2,  and  allow  C  to  change  in  value. 
D2    r.p.m.i 


D\     r.p.m.2 

2 

i  =  — tan  OL\\ 

i 

.'.  tan  «i  =  i; 
Trial  L\  =  nD\  sin  a\ 


(3) 


Trial  Lz^nD-z  cos  a\ 


402  MACHINE  DESIGN. 


But  p  =  -j  —  and  N\  has  been  assumed  =  10; 
-^i 

IOX7T 

=  6.63. 


The  nearest  standard  diametral  pitch  to  this  is  7.     Selecting 
p  =  J,  with  NI  =  IO, 

Actual,  or  corrected,  Li  =  NiP  =  — -1-, 

P 

.*.  Correct  —  =  — *=« — =1.420 

7T  p  7 

Actual,  or  corrected,  L2  =  N2P  =  N2-, 

P 

L2    N2     20 
.'.  Correct  —  =  — =_=2.8c8. 

7T  p  7 

correct  LI  i  420 

Correct  ^I=T^T> w    =™3.I95". 

Driver  gear-blank  diameter  =  Z?i  +  —  =  3. 195"  +  . 286"  =3. 481" 

P 

correct  L2  2.8^8 

Correct  D2=~  — , (8) 


.8944 


ty 

Follower  gear-blank  diameter  =  D2  -\ — =3. 195"  +  .2 

P 

To  select  the  cutter  of  the  7  diametral  pitch  set: 

Driver      N0=  .     *  .  = -  =  m.8. 

sm-^ai     .4472^ 

calling  for  B.  &  S.  involute  cutter  No.  2. 

_,  „  ,r        N2          20 

Follower  A^o  = = =  27  oc 

sin3o:2     .89442       />y5' 

calling  for  B.  &  S.  involute  cutter  No.  4. 


TOOTHED   WHEELS   OR   GEARS.  403 

To  determine  the  lead  of  tooth  helix,  in  order  to  select  the 
corresponding  gear  set  of  the  milling  machine: 

Driver  lead  =  7r.Di  tan  a.\ 


calling  for  gears  86,  48,  28,  and  100  in  B.  &  S.  milling  machine. 

Follower  lead  =  7rjD2  cot  «i 
=  7rX3. 195X2 
=  20.075", 

calling  for  gears  86,  24,  56,  and  100  in  B.  &  S.  milling  machine. 
Summary  for  modified  distance  between  centers: 

Driver.  Follower. 

Pitch  diameter,     1)1  =  3.195"  ^2  =  3.195" 

Gear-blank  diameter  =  3. 481"  Blank  diameter  =  3. 481" 
Number  of  teeth,  N\  =  10  N2=2o 

Helix  angle,  0:1  =  26°  34'  0:2  =  63°  26' 

Diametral  pitch,      p  =  7  p  =  7 

Cutter,  involute,  No.  2  Cutter  No.  4 

Lead  of  tooth  helix    =5.018"  Lead  =20.075" 

Gears,  86,  48,  28,  100  Gears,  86,  24,  56,  100 


-^-  =3-195". 

Second  Solution. 

Taking  the  same  data  and  assuming  that  the  center  dis- 
tance remains  fixed  at  3.375",  it  is  still  desired  to  have  D\ 
and  DZ  as  nearly  equal  as  possible. 

If  NI=IO,  N2=2o,  and  p  =  *j  it  is  fixed  that 


and  £2=^X2.858. 


404  MACHINE  DESIGN. 

T  /~* 

From  equation  (12)   i+—  tan  a\  =  —  sin  a\9 
LI  LI 

n 

_^75_ 
1.429 

or,  1  +  2  tan  0:1  =  4.724  sin  «i. 

The  next  step  is  to  substitute  trial  values  of  a\  until  a  value 
is  found  which  will  make  the  two  sides  of  the  equation  equal. 
If  the  right-hand  member  comes  out  greater  than  the  left  the 
trial  value  of  a\  is  too  large;  if  the  left-hand  member  comes 
out  greater  than  the  right  the  trial  value  of  a\  is  too  small. 

Starting  with  a  trial  value  of  0:1  =  26°  34',  from  the  first 
solution,  it  is  found  to  be  too  large.  A  few  trials  lead  to  the 
value  of  23°  5'  for  a\  which  gives 

i  +  2  X  .42619  =  4.724  X  .39207, 
.*.  1.852=1.852. 

Therefore  the  correct  0:1  =  23°  5',  and 

correct  0:2  =  90°— 0:1  =  66°  55' 

correct  L\  1.420 

Correct  D\  =  — : —  —  =  3.645". 

TT  sin  ai  (correct)     .3921 

correct  L2  2.858 

Correct  D2=-  — =-      -  =  3.106". 

TT  cos  a  i  (correct)     .  9 1 993 

2 

Driver  gear-blank  diameter     =  D\  +  — = 3 .645 "  + .  286  =  3 .93 1 ". 

2 

Follower  gear-blank  diameter  =  D2  +  -  =  3. 106"  +  .286"  =  3.392"., 


Driver      ^0=  —  —=165.9, 

3-  3 


Ni          10 
sin3o-i 

calling  for  B.  &  S.  involute  cutter  No.  i. 


TOOTHED   WHEELS    OR   GEARS. 


Follower  NQ  = 


=  25.1 


i      .9199 
calling  for  B.  &  S.  involute  cutter  No.  5. 

Driver  tooth-helix  lead  =  7rZ>i  tan  OL\ 


405 


calling  for  B.  &  S.  gear-set,  48,  64,  56,  86. 

Follower  tooth  -helix  lead  =  7r£>2  cot  a\ 

=  71X3.  106X2.3463 
=  22.9-, 

calling  for  B.  &  S.  gear-set,  72,  44,  56,  40. 

Summary  for  fixed  distance  between  centers: 

Driver. 

Pitch  diameter,     DI  =  3.645" 
Gear-blank  diameter  =  3.  931" 
Number  of  teeth,  NI  =  10 
Helix  angle  0:1  =  23°  5' 

Diametral  pitch       p  =  7 
Cutter,  involute  No.  i 
Lead  of  tooth  helix    =4.88" 
Gears,  48,  64,  56,  86 


Follower. 

#2  =  3.106" 

Blank  diameter  =  3.  392" 
N%  =  20 
a2  =  66°  55' 
p  =  7 

Cutter  No.  5 
Lead  =22.9" 

Gears,  72,  44,  56,  40 


n 

c= 


=3-375 


224.  Spiral  Gears  with  Axes  at  any  Angle,  /8.  —  Fig.  228^ 
shows  a  plan  view  of  such  a  pair  of  gears,  and  also  a  view  of  the 
gears  developed  in  the  tangent  plane. 

From  the  latter  it  is  evident  that  a  motion  ba  of  the  driver 
in  its  direction  of  rotation  must  induce  a  motion  be  of  the 


4o6 


MACHINE  DESIGN. 


follower  in  its  direction  of  rotation.     This  establishes  the  fact 
that 

circumferential  velocity  of  follower     be 
circumferential  velocity  of  driver      ba' 


\ 


FIG.  22BE. 

Consider  the  triangle  abc.    Angle  cab=ait  angle  acb 

be     sin  ai 
ba     sin  0:2  " 

circumferential  velocity  of  follower     7rZ>2  r.p.m.2 
circumferential  velocity  of  driver      izD\  r.p.m.i' 

7tD2  r.p.m.2     sin«i 
TrDir.p.m.i     sin  a.% 

D%    r.p.m.i  sin  «i 


D\     r.p.m.2  sin  a^ 


TOOTHED    WHEELS  OR   GEARS.  407 

r.p.m.2    DI  sin  «i 

or,  =  7^ — : \2) 

r.p.m.i     DI  sin  a2 


/.  D2=2C-Dl (3) 

Substitute  in  (i) 

2C—Di     r.p.m.i  sin  a.\ 
D\         r.p.m.2  sin  0.% 


.p.m.2  sin  a2 


r.p.m.2  sin  a2 

2C 


(4) 


Because  ai+/?+a2=i8o0,  sin  a2  =  sin  (/?+«i)   and   (4)   may 
also  be  written 


. 

r.p.m.i  sin  OL\ 

r.p.m.2  sin  a2 


. 
r.p.m.i  sin  OL\ 


(5) 


r.p.m.2  sin 
[NOTE. — Equations   (4)   or   (5)   and   (3)   give  us  all  possible 

solutions  for  anv  value  of  C  and  -^ — — ,  just  as  when  the  axes 

r.p.m.2   " 

were  at  90°.     In  fact  (5)  reduces  to  the  form  used  in  that  case 
when  /?=90°,  for 

2C 2C 

r.p.m.i  sin  a\  r.p.m.i  sin  a\ 


r.p.m.2  sin  (90°  + ai)  r.p.m.2  cos  a\ 

2C 


r.p.m.i 

—   -tanai. 
r.p.m.2 


408  MACHINE  DESIGN. 

It  can  be  shown  (Reuleaux's  "  Constructor ")  that  the 
sliding  velocity  (along  ac)  of  the  teeth  upon  each  other  is  least 
when  0:1  =  0:2;  .*.  whenever  possible  these  values  should  be 
given  «i  and  0:2-  That  is,  ai  =  a2  =  ^(iSo°  —  /?). 

We  have  already  seen  that  when  ft =90°,  a  1==  0:2  =  45°  is  the 
most  efficient  angle  of  helix. 

It  must  be  borne  in  mind  that  these  values  of  «i,  o:2,  DI, 
and  D2  may  give  us  impractical  values  for  normal  pitch  and 
that,  in  consequence,  the  values  may  have  to  be  modified  to  get 
a  normal  helix  length  which  will  give  an  even  number  of  teeth 
of  stock  size.] 

As  in  spiral  gears  with  axes  at  90°  we  have 

Li  =  xDi  sin  ai,        (6) 

L2  =  nD2  sin  0:2 (7) 


But  Li  =  NiP=Ni-. 

P 


D= 


1             —                    (S 

(_Ni  esc  ai\ 

n  sin  0:1     p  sin  ai* 
pD\  sin  a.i\       

(             P         ) 

(Q) 

N2 

/       N2  CSC  «2\ 

p  sin  o:2' 

1      p    ) 

(ill 

CSC  ai  +      *  CSC  a'2 

—  .      .     .     .     (12) 


The  foregoing  equations  may  be  used  to  get  the  practical 
solution. 

225.  Problem.  —  The   following  problem  will  illustrate   the 
method.     Compute  a  pair  of  spiral  gears, 

Shaft  angle,  /?=4O°, 


TOOTHED    WHEELS   OR   GE4RS.  409 

Exact  center  distance  =  3"  (not  to  be  changed), 
r.p.m.  1  =  400, 

r.p.m.2  =  300, 
p=io. 

Solution. 


-40°  -=140°, 


140° 


Try  fl^at**— «7<r, 

From  equation  (4), 


r.p.m.i  sin  «i  r.p.m.i  4     2.333 

•  -    -  ;:  -      JH  -- 

r.p.m.2  sin  0:2          r.p.m.2          3 


From  (9),  Ni  =  pDi  sin 


From  (10),        N2  =  pD2  sin  «2, 

AT2=  10X3.43  X.94  =  32-I3- 

300 
But  NI  and  ^2  must  be  whole  numbers  in  ratio  of  -  ,  or 

24  and  32,  respectively. 

Use  these  values  of  NI  and  N2. 
Substitute  in  (12) 


c-/vjac 

2/> 

C  1.064 +  ^2X1. 064 

=  2.979. 


2P 

24X  1.064  +  32  X  1.064 


2X10 


4io 


MACHINE  DESIGN. 


It  is  evident  that  new  values  of  a\  and  a2  will  have  to  be  tried 
to  make  this  equation  an  identity.  It  is  necessary  to  take  trial 
values  of  0.1  and  a2,  remembering  that  the  sums  of  «i  and  a2 
must  always  be  180°— /?  (=140°  in  this  case). 

Try  0:1  =  74°,  a2  =  66°.     Substitute  in  (12) 

24X1.0403+32X1.0946 


which  will  answer. 


^  icscai      24X1.0403 

From    (8),  correct  D\  =  --  =—  -  -  —  =  2.497". 

p  10 

._  *      N2csca2     32X1.0946 
From  (10),  correct  D2  =  -  =-  --  ^~=3-5°3  • 

2 

Driver  gear-blank  diameter=Z>iH  —  =  2.  497^  +  .  2"  =  2.  697". 
Follower  gear-blank  diameter  =  D2  +  -  =  3.  507"  +  .  2"  =  3.  703". 

Lead  of  tooth-helix  and  cutter  to  be  used  are  found  as  in  spiral 
gears  with  axes  at  90°.  For  follower,  however,  use  nDz  tan  0:2, 
not  7rZ>2  cot  ai. 

From  the  nature  of  hyperboloidal  wheels,  two  solutions  are 
always  possible,  depending  upon  whether  /?  or  its  supplement 
be  taken  in  determining  the  line  of  contact  of  the  hyperboloids. 
In  this  problem  it  is  evidently  just  as  proper  to  consider  the 
shaft  angle  to  be  140°  as  40°.  See  Fig.  22&F. 

Here 


=  i8o°-i40°, 


=  40°. 


TOOTHED    WHEELS  OR   GEARS. 


1  + 


r.p.m.i  sin  a\ 


3-51", 


r.p.m.2  sin 
£2=2.49". 


V 


o        .— H  I 

'**  J 

>-— H     ?/ 


SPIRALS  > 

DRIVER  LEFT  HAND       * 
FOLLOWER  RIGHT  HAND/ 


X 


Axis  of 
'  Follower 


FIG.  228F. 


Trial  NI  =  17.55,  ^2=23.4  (from  (9)  and  (10)), 
Correct,  JVi  =  18,     N2  =  24. 


C  = 


•*•  3 


showing  necessity  of  modifying  «i  and  «2. 
Try  71°  15'  and  31°  15'  for  ai  and  a2, 

18X1.9276  +  24X1.056 
o  =  --  .  -  —  —  3.002. 

20 

Hence  these  values  of  «i  and  o:2  will  answer. 


Correct  Di 


CSC 


3-47 


N2  esc  a  2 
Correct  D2=-  -  =  2.53". 


412  MACHINE  DESIGN. 

[It  is  to  be  noted  that  in  the  two  cases  the  spirals  have  differ- 
ent relations.  In  the  first  case  where  the  spirals  are  both  of 
the  same  hand,  /?+  a:  1  +  0:2=180°,  i.e., 


In  the  second  case,  where  the  spirals  are  of  opposite  hands, 


.e.,  a2  —  ai 

The  topic  of  direction  of  spirals  and  direction  of  rotation 
is  well  treated  in  the  American  Machinist,  Oct.  n,  1906.] 

226.  Worm-gearing.  —  When  the  angle  between  the  shafts  is 
made  equal  to  90°,  and  one  gear  has  only  one,  two,  three,  or  four 
threads,  it  becomes  a  special  case  of  spiral  gearing  known  as 
Worm-gearing.  In  this  special  case  the  gear  with  a  few  threads 
is  called  the  worm,  while  the  other  gear,  which  is  still  a  many- 
threaded  screw,  is  called  the  worm-wheel.  If  a  section  of  a  worm 
and  worm-wheel  be  made  on  a  plane  passing  through  the  axis 
of  the  worm,  and  normal  to  the  axis  of  the  worm-wheel,  the  form 
of  the  teeth  will  be  the  same  as  that  of  a  rack  and  pinion;  in 
fact  the  worm,  if  moved  parallel  to  its  axis,  would  transmit  rotary 
motion  to  the  worm-wheel.  From  the  consideration  of  racks 
and  pinions  it  follows  that  if  the  involute  system  is  used,  the  sides 
of  the  worm-teeth  will  be  straight  lines.  This  simplifies  the  cutting 
of  the  worm,  because  a  tool  may  be  used  capable  of  being  sharpened 
without  special  methods.  If  the  addendum  equals  the  reciprocal 
of  the  diametral  pitch,  it  follows  from  the  interference  formula 
that  a  pressure  angle  of  75°  30'  (14!°  involute  system)  calls  for 
at  least  32  teeth  on  the  wheel.  For  wheels  with  a  smaller  number 
of  teeth  than  this,  the  length  of  worm  addendum  must  be  shortened 
or  the  pressure  angle  made  smaller  (i.e.,  tooth  angle  increased). 
If  the  worm-wheel  were  only  a  thin  plate  the  teeth  would  be 
formed  like  those  of  a  spur-gear  of  the  same  pitch  and  diameter. 
But  since  the  worm-wheel  must  have  greater  thickness,  and 
since  all  other  sections  parallel  to  that  through  the  axis  of  the 


TOOTHED   WHEELS  OR   GEARS. 


413 


worm,  as  CD  and  AB,  Fig.  229,  show  a  different  form  and 
location  of  tooth,  it  is  necessary  to  make  the  teeth  of  the  worm- 
wheel  different  from  those  of  a  spur-gear,  if  there  is  to  be  con- 
tact between  the  worm  and  worm-wheel  anywhere  except  in 
the  plane  EF,  Fig.  229.  This  is  accomplished  in  practice  as 
follows :  A  duplicate  of  the  worm  is  made  of  tool  steel,  and  "  flutes  " 
are  cut  in  it  parallel  to  the  axis,  thus  making  it  into  a  cutter, 
which  is  tempered.  It  is  then  mounted  in  a  frame  in  the  same 
relation  to  the  worm-wheel  that  the  worm  is  to  have  when  they 
are  finished  and  in  position  for  working.  The  distance  between 
centers,  however,  is  somewhat  greater,  and  is  capable  of  being 
gradually  reduced.  Both  are  then  rotated  with  the  required 
velocity  ratio  by  means  of  gearing  properly  arranged,  and  the 
cutter  or  "hob  "  is  fed  against  the  worm-wheel  till  the  distance 
between  centers  becomes  the  required  value.  The  teeth  of  the 

ACE 


FIG.  229  FIG.  230. 

worm-wheel  are  "roughed  out  "  before  they  are  "bobbed."  By 
the  above  method  the  worm  is  made  to  cut  its  own  worm- 
wheel.*  A  more  modern  method  is.  to  use  a  taper  hob,  fed 
axially.  The  worm  itself  is  milled,  not  turned,  in  many  cases. 

The  worm  may  have  the  basic  form  corresponding  to  the 
root  circle  of  the  central  plane  of  the  wheel.  This  is  known 
in  America  as  the  Hindley  worm.  It  gives  more  teeth  in  simul- 
taneous engagement  than  the  ordinary  cylindrical  worm. 

Fig.  230  represents  part  of  the  half  section  of  a  worm.  If  it 
is  a  single  worm  the  thread  A,  in  going  once  around,  comes  to 


*This  subject  is  fully  treated  in  Unwin's  "Elements  of  Machine  Design," 
and  in  Brown  and  Sharpe's  "Treatise  on  Gearing." 


414  MACHINE  DESIGN. 

B ;  twice  around  to  C,  and  so  on.  If  it  is  a  double  worm  the  thread 
A,  in  going  once  around,  comes  to  C,  while  there  is  an  inter- 
mediate thread,  B.  It  follows  that  if  the  single  worm  turns 
through  one  revolution  it  will  push  one  tooth  of  the  worm-^heel 
with  which  it  engages  past  the  line  of  centers;  while  the  double 
worm  will  push  two  teeth  of  the  worm-wheel  past  the  line  of 
centers.  The  single  worm,  therefore,  must  make  as  many  revolu- 
tions as  there  are  teeth  in  the  worm-wheel,  in  order  to  cause 
one  revolution  of  the  worm-wheel;  while  for'  the  same  result 
the  double  worm  only  needs  to  make  half  as  many  revolutions. 
The  ratio  of  the  angular  velocity  of  a  single  worm  to  that  of 

n 
the  worm-wheel  with  which  it  engages  is=— ,  in  which  n  equals 

the  number  of  teeth  in  the  worm-wheel.     For  the  double  worm 

n 

this   ratio  is   — . 
2 

Worm-gearing  is  particularly  well  adapted  for  use  where  it 
is  necessary  to  get  a  high  velocity  ratio  in  limited  space. 

The  lead  of  a  worm  is  measured  parallel  to  the  axis  of  rota- 
tion. The  lead  of  a  single  worm  is  P,  Fig.  230.  It  is  equal 
to  the  circular  pitch  of  the  worm-wheel.  The  lead  of  a  double 
worm  is  PI  =  2P  =  2  X  circular  pitch  of  the  worm-wheel, 

227.  Design  of  Worm-gears.— All  spiral  gears  are  forms  of 
screw  transmission  and  the  formulae  for  efficiency,  etc.,  developed 
under  c,  sec.  98,  in  the  chapter  on  Screws,  apply  to  them  di- 
rectly. 

Three  points  are  to  be  carefully  considered  in  the  design  of 
worms  and  wheels: 

i.  Speed  of  rubbing.  This  is  the  velocity  in  feet  per  minute 
of  a  point  on  the  pitch  line  of  the  worm.  The  best  efficiencies 
are  obtained  when  this  is  about  200  feet  per  minute.  When 
it  exceeds  300  feet  there  is  increasing  danger  of  cutting,  and 
the  pressure  on  the  tee^th  must  be  correspondingly  reduced 


TOOTHED   WHEELS   OR   GEARS.  415 

At  high  speeds    (say  1000  feet)   only  very  light  pressure  can  be 
sustained  without  abrasion  unless  there  is  bath  lubrication. 

2.  Pressure  on  teeth.     This  depends  on  the  speed  and  on  the 

angle  of  helix. 

3.  Angle  oj  helix.     From  the  formula  for  screw  efficiency  we 
have  seen  that  this  should  be  made  as  great  as  convenient  provided 
it  does  not  exceed  45°.     Practical  conditions  make  it  impossible 
to  use  the  highest  values,  but  20°  gives  very  excellent  results.    It 
should  never  be  less  than  15°  for  fair  efficiency. 

Oil -bath  lubrication  should  be  used  wherever  possible;  failing 
this,  a  heavy  mixture  of  graphite  and  oil  has  been  found  satis- 
factory. The  following  table,  based  on  Professor  Stribeck's 
experiments,*  applies  to  a  20°  angle  of  helix  and  oil-bath  lubri- 
cation, using  a  hardened-steel  worm  and  phosphor-bronze  wheel. 

TABLE  XXXIX. 


Rubbing  velocity  in  feet  per  minute  

200 

300 

400 

500 

600 

Allowable  pressure  for  maximum  efficiency 

and  continuous  operation  in  pounds  

35oo 

2700 

1850 

1250 

IOOO 

About  60  per  cent  heavier  loads  than  these  were  borne,  but 
at  a  loss  in  efficiency,  under  continuous  operation.  For  discon- 
tinuous operation,  very  much  heavier  loads  still  are  permissible- 
It  is  largely  a  question  of  not  allowing  the  lubricant  to  reach 
a  temperature  at  which  the  pressure  will  rupture  the  film  and 
permit  metallic  contact. 

This  may  be  taken  as  a  guide.  When  the  angle  is  greater 
than  20°  the  values  of  the  pressure  may  be  slightly  increased. 
When  the  angle  is  less  than  20°  they  should  be  rapidly  diminished ; 
thus  for  10°  use  only  one  half  the  value  given. 

There  is  ordinarily  little  need  to  examine  the  strength  of 

*  Zeitschrift  d.  Vereins  deutscher  Ingenieure,  1897;  also  1898. 

I 


416  MACHINE    DESIGN. 

worm  or  wheel  teeth.  The  permissible  load  (turning  force  at 
wheel  pitch  circumference)  is  limited  by  questions  of  number 
of  teeth  in  simultaneous  contact,  form  of  tooth  profile,  nature 
of  lubricant  and  its  method  of  application,  allowable  rise  in 
temperature,  etc.  It  is  well  to  check  for  the  twisting  strength 
of  the  core  of  the  worm. 

Bach  and  Roser*  give  the  following  formula  for  soft  sceel 
worm  engaging  a  bronze  wheel  (helix  angle  17°  34';  15°  in- 
volute teeth),  flooded  lubrication. 

W==KPb; 


21,476 


W  =  tangential  force  at  pitch  circumference  of  wheel,  pounds  ; 
P  =  circular  pitch,  inches; 

Z>  =  arc  length   of   root   of  worm-wheel   teeth,    inches    (EFf 

Fig.  231); 
/i  =  temperature  of  oil  bath,  deg.  Fah.; 

/  =  temperature  of  air,  deg.  Fah.; 
V  =  pitch  velocity  of  worm,  feet  per  minute. 

This  is  a  better  formula  than  Stribeck's  value  of 

W  =  3$6Pb,  for  cast-iron  wheel, 

=  56QP6,  for  phosphor  bronze  wheel, 
since  it  is  more  general. 

*  Z.  d.  V.  d.  I.,  1903.     Prof.  Kenerson's  experiments,  Trans.  A.  S.  M.  E.,  Vol. 
34,  indicate  values  of  K  about  four  times  as  great. 


TOOTHED   WHEELS  OR   GEARS.  417 

Cast-iron  worms  and  wheels  will  run  satisfactorily  under 
certain  conditions,*  but  a  worm  of  a  steel  which  case-hardens 
well,  engaging  a  wheel  of  best  phosphor  bronze,  seems  to  give 
best  service.  Properly  designed  and  installed  worms  and  wheels 
show  high  efficiencies.!  Particular  care  must  be  taken  with 
the  thrust  bearings. 

Since  worms  and  wheels  are  simply  spiral  gears  in  which 
one  of  the  gears  has  a  very  few  teeth,  all  of  the  general  formulae, 
relating  to  DI,  D2,  «i,  a2,  p,  etc.,  developed  in  the  preceding 
sections,  are  directly  applicable  in  their  design.  However,  as 
worms  are  frequently  cut  in  a  lathe  (like  screw  threads)  and  worm- 
wheels  hobbed,  it  is  the  axial  pitch  of  the  worm,  equal  to  the 
circumferential  pitch  of  the  wheel,  which  is  of  importance  rather 
than  the  normal  pitch.  This  "  lead  "  must  then  have  a  value 
obtainable  with  the  screw-cutting  gearing  of  the  lathe.  It  is 
therefore  practically  more  convenient  in  the  design  of  worms 
and  wheels  to  follow  the  method  illustrated  by  the  following 
problem : 

228.  Problem. — Two  shafts  about  10  inches  apart  and  at  a 
right  angle  with  one  another  are  to  have  a  velocity  ratio  of  20  to  i. 
The  worm-shaft  makes  300  revolutions  per  minute. 

Since  the  velocity  ratio  is  20  to  i,  the  wheel  will  have  to  have 
20,  40,  or  60  teeth,  depending  upon  whether  the  worm  is  single, - 
double-,  or  triple-threaded. 

If  the  shafts  are  10  inches  apart  the  greatest  allowable  pitch 
radius  of  the  wheel  will  not  be  far  from  8  inches;  50  inches  may 
be  taken  as  a  trial  pitch  circumference  of  the  wheel. 

With  a  single-thread  worm  this  will  give  a  circular  pitch  of 
f#  =  2\  inches.  With  a  double  thread  the  circular  pitch  would 
be  |$=i  J  inches. 

*  Stribeck.    Z.  d.  V.  d.  I.,  1898. 

t  See  further  Kenerson,  Trans.  A.  S.  M.  E.,  Vol.  XXXIV.  Also  Bruce,  Proc. 
Inst.  M.  E.,  1905. 


418  MACHINE  DESIGN. 

In  any  case  the  rise  of  the  pitch  helix  of  the  worm  will  be 
2j  inches  for  one  revolution. 

This  value  must  always  be  such  that  the  thread  may  be  cut  in 
an  ordinary  lathe. 

If  it  is  required  that  the  helix  angle,  a,  be  20°,  then  the 
pitch  circumference  of  the  worm  must  be  such  that 


o  * 

°"  '         pitch  circumference  of  worm; 

.*.  pitch  circumference  of  worm  =  —    —  =6.87  inches. 

0.364 

6.87 
Pitch  diameter  of  worm  =  -  =  2.2  inches. 


Pitch  diameter  of  wheel  =  —  =  1  5  .  88  inches. 

Actual  distance  between  shafts  =  —  —=9.04  inches.- 

The  question  now  arises  whether  2.2  inches  is  a  great  enough 
pitch  diameter  for  the  worm.  If  the  thread  is  single  the  pitch 
'=2.5  inches  and  the  corresponding  dedendum  =0.92  inch. 

Twice  this  dedendum  =  1.84  inches,  which  subtracted  from 
2.2  inches  would  only  leave  a  central  solid  core  of  0.36  inch  diam- 
eter for  the  worm.  It  is  obvious  without  computation  that  this 
would  not  sustain  the  torsional  moment.  If  the  double  thread 
were  used  the  central  core  would  have  a  diameter  of  1.28  inches. 

For  each  revolution  of  the  worm  the  length  of  the  path  of 
the  point  of  contact  or  the  distance  rubbed  over  equals  the  helix 
length  on  the  pitch  line.  This  is  the  hypothenuse  of  a  right- 
angle  triangle  whose  base  =6.87  inches  and  whose  altitude  = 
2.5  inches,  or  7.3  inches. 

At  300  revolutions  per  minute  the  distance  rubbed  through  in 


TOOTHED    WHEELS  OR   GEARS.]  419 

feet  per  minute  =  300  X  —  =  182  feet.     At  182  feet  the  allowable 

pressure,  W,  between  the  teeth  may  equal  3500  Ibs.,  assuming 
bath  lubrication,  a  steel  worm,  and  a  bronze  wheel.  This  is  the 
pressure  applied  at  the  circumference  of  the  worm-wheel  in  the 
direction  of  the  axis  of  the  worm.  The  total  work  done  on  the 
worm-wheel  in  foot-pounds  per  minute  will  equal  W  multiplied 
by  the  pitch  velocity  of  the  wheel  in  feet  per  minute. 

This  wheel  makes  3Aa  =  J5  revolutions  per  minute  and  its 
pitch  circumference  ={•  f  feet,  hence  its  pitch  velocity  =  i  SXinr 
=  62.5  feet  per  minute. 

3500X62.5  =218,750  ft.-lbs.  =6.63  H.P. 

This  same  amount  of  energy  is  transmitted  through  the  worm. 
The  twisting  moment  on  the  shaft  =Fr,  where  r  equals  the  pitch 
radius  of  the  worm.  F=  energy  transmitted  -f-  the  velocity  of 
the  point  of  application  of  the  force. 


21 


Pr=  1274X1.  i  =  1401  in.-lbs. 

To  resist  this  there  is  a  circular  section  whose  strength  is 

fxri3 
represented  by/s  -  . 

r\  =  radius  of  core  of  worm  =  0.64  inch  ; 
/*  =  unit  stress  in  outer  fiber; 
Fr      1401 


2 

This  is  a  safe  value  for  steel.  Therefore  the  double-threaded 
worm  will  be  used  and  the  wheel  will  have  40  teeth  of  ij  inches 
circular  pitch. 


420  MACHINE    DESIGN. 

Had  the  distance  between  the  shafts  been  fixed  at  10  inches 
the  helix  angle  could  not  have  been  assumed  but  must  have  been 
calculated. 

The  pitch  radius  of  worm  would  have  been 


Pitch  circumference  =  12.94  inches. 


2. 


. 

Tangent  of  index  angle  =  —    -  =  o.  1032. 

12.94 

/.  a  =11°  nearly. 

With  the  center  distance  fixed  at  10  inches,  the  helix  angle 
need  not  necessarily  be  as  low  ac  11°;  provided,  of  course,  that 
the  axial  pitch  of  the  worm  may  be  changed  to  some  other  value 
greater  than  2.5".  As  a  check  on  the  final  results  the  general 
formulae  of  the  preceding  sections  may  be  applied  to  the  values 
obtained  by  the  method  here  followed. 

When  the  worm  and  worm-wheel  are  determined,  a  working 
drawing  may  be  made  as  follows:  Draw  AB,  Fig.  231,  the  axis 
of  the  worm-wheel,  and  locate  O,  the  projection  of  the  axis  of 
the  worm,  and  P,  the  pitch-point.  With  O  as  the  center  draw  the 
pitch,  full  depth,  and  addendum  circles,  G,  H,  and  K\  also 
the  arcs  CD  and  EF,  bounding  the  tops  of  the  teeth  and  the 
bottoms  of  the  spaces  of  the  worm-wheel.  Make  the  angle  0  =  90°. 
Below  EF  lay  off  a  proper  thickness  of  metal  to  support  the  teeth 
and  join  this  by  the  web  LM  to  the  hub  N.  The  tooth  outlines 
in  the  other  sectional  view  are  drawn  exactly  as  for  an  involute 
rack  and  pinion.  Full  views  might  be  drawn,  but  they  involve 
difficulties  of  construction,  and  do  not  give  any  additional  infor- 
mation to  the  workman.  The  drawing  should  contain  a  clear 
statement  of  the  size  and  form  of  the  worm  tooth,  the  lead, 
whether  the  worm  is  single,  double,  triple,  or  quadruple  threaded, 


TOOTHED    U/HEELS  OR  GEARS. 


421 


the  number  of  teeth  of  the  wheel,  and  its  helix  angle,  in  addition 
to  all  ordinary  dimensions. 


FIG.  231. 

229.  Compound  Spur-gear  Chains. — Spur-gear  chains  may 
be  compound,  i.e.,  they  may  contain  links  which  carry  more  than 
two  elements.  Thus  in  Fig.  232  the  links  a  and  d  each  carry 
three  elements.  In  the  latter  case  the  teeth  of  d  must  be  counted 
as  two  elements,  because  by  means  of  them  d  is  paired  with  both 
b  and  c.  In  the  case  of  the  three-link  spur-gear  chain,  Fig.  197, 
the  wheels  b  and  c  meshed  with  each  other,  and  a  point  in  the 
pitch  circle  of  c  moved  with  the  same  linear  velocity  as  a  point  in  the 
pitch  circle  of  6,  but  in  the  opposite  sense.  In  Fig.  232  points  in 
all  the  pitch  circles  have  the  same  linear  velocity,  since  the  motion 
is  equivalent  to  rolling  together  of  the  pitch  circles  without  slip- 
ping; but  c  and  b  now  rotate  in  the  same  direction.  Hence  it 
is  seen  that  the  introduction  of  the  wheel  d  has  reversed  the 
direction  of  rotation,  without  changing  the  velocity  ratio.  The 


422 


MACHINE  DESIGN. 


size  of  the  wheel,  d,  which  is  called  an  "idler,"  has  no  effect  upon 
the  motion  of  c  and  b.  It  simply  receives  upon  its  pitch  circle 
a  certain  linear  velocity  from  c}  and  transmits  it  unchanged 
to  b.  Hence  the  insertion  of  any  number  of  idlers  does  not 
affect  the  velocity  ratio  of  c  to  6,  but  each  added  idler  reverses 
the  direction  of  the  motion.  Thus,  with  an  odd  number  of 
idlers,  c  and  b  will  rotate  in  the  same  direction;  and  with  an 
even  number  of  idlers  c  and  b  will  rotate  in  opposite  directions. 

If  parallel  lines  be  drawn  through  the  centers  of  rotation  of 
a  pair  of  gears,  and  if  distances  be  laid  off  from  the  centers  on 
these  lines  inversely  proportional  to  the  angular  velocities  of  the 
gears,  then  a  line  joining  the  points  so  determined  will  cut  the 
line  of  centers  in  a  point  which  is  the  centre  of  the  gears.  In  Fig. 
232,  since  the  rotation  is  in  the  same  direction,  the  lines  have  to 


FIG.  232. 


FIG.  233. 


be  laid  off  on  the  same  side  of  the  line  of  centers.  The  pitch 
radii  are  inversely  proportional  to  the  angular  velocities  of  the 
gears,  and  hence  it  is  only  necessary  to  draw  a  tangent  to  the  pitch 
circles  of  b  and  c,  and  the  intersection  of  this  line  with  the  line  of 
centers  is  the  centro,  be,  of  c  and  b.  The  centrodes  of  c  and  b  are 
Ci  and  61,  circles  through  the  point  be,  about  the  centers  of  c  and  b. 
Obviously  this  four-link  mechanism  may  be  replaced  by  a  three- 
link  mechanism,  in  which  Ci  is  an  annular  wheel  meshing  with  a 
pinion  61.  The  four  link  mechanism  is  more  compact,  however, 
and  usually  more  convenient  in  practice. 

The  other  principal  form  of    spur-gear  chain  is  shown  in 


TOOTHED   WHEELS   OR   GEARS. 


42* 


Fig.  233.  The  wheel  d  has  two  sets  of  teeth  of  different  pitch 
diameter,  one  pairing  with  c  and  the  other  with  b.  The  point" 
bd  now  has  a  different  linear  velocity  from  cd,  greater  or  less  in 
proportion  to  the  ratio  of  the  radii  of  those  points.  The  angular 
velocity  ratio  may  be  obtained  as  follows: 

angular  velocity  d  __  C  .  .  .  cd 
angular  velocity  c     D  ...  cd  ' 

.  angular  velocity  b  _  D  .  .  .  bd 

angular  velocity  d      B  .  .  .  bd' 

angular  velocity  b      C  .  .  .  cdxD  .  .  .  bd 


angular  velocity  c 

The  numerator  of  the  last  term  consists  of  the  product  of  the 
radii  of  the  "  followers,"  and  the  denominator  consists  of  the 
product  of  the  radii  of  the  "drivers."  The  diameters  or  numbers 
of  teeth  could  be  substituted  for  the  radii. 

In  general,  the  angular  velocity  of  the  first  driver  is  to  the 
angular  velocity  of  the  last  follower  as  the  product  of  the  number 
of  teeth  of  the  followers  is  to  the  product  of  the  number  of  teeth 
of  the  drivers.  This  applies  equally  well  to  compound  spur-gear 
trains  that  have  more  than  three  axes.*  Therefore,  in  any  spur- 
gear  chain  the  velocity  ratio  equals  the  product  of  the  number  of 
teeth  in  the  followers  divided  by  the  product  of  the  number  of 
teeth  in  the  drivers.  The  direction  of  rotation  is  reversed  if  the 
number  of  intermediate  axes  is  even,  and  is  not  reversed  if  the 
number  is  odd.  If  the  train  includes  annular  gears  their  axes 
would  be  omitted  from  the  number,  because  annular  gears  do 
not  reverse  the  direction  of  rotation. 

A  common  modification  of  the  chain  of  Fig.  233  is  shown  in 
Fig.  233  A.     Here  the  axis  of  the  gear  c  is  made  to  coincide  with 

*  Epicyclic  trains  excepted. 


424 


MACHINE  DESIGN. 


the  axis  of  b,  and  the  mechanism  is  known  as  a  reverted  gear  train. 
Probably  the  best  known  application  of  this  mechanism  is  that  of 
the  backgearing  of  the  ordinary  engine  lathe.  The  velocity  ratio 
of  c  and  b  is,  of  course,  not  altered  by  having  their  axes  coincide, 
and  it  is  equally  evident  that  one  of  them  only  may  be  keyed  to 
the  shaft  while  the  other  is  free  to  rotate  on  it. 


FIG.  233  A.  FIG.  2336. 

230.  Epicyclic  Gearing. — In  the  gear  trains  of  the  preceding 
sections,  the  velocity  ratios  have  been  studied  with  reference  to 
a  fixed  member  to  which  each  gear  is  attached  by  a  turning  pair. 
Fig.  233  B  illustrates  such  a  simple  chain  of  three  links,  a,  b,  and 
c.  Considering  a  as  fixed  it  is  evident  that,  if  c  makes  m  turns 
per  minute  relatively  to  a,  causing  b  to  make  n  turns  per  minute 

relatively  to  a,  for  one  turn  of  c  relatively  to  #,  b  will  make  —  turns 

relatively  to  a.     The  ratio  —  is  called  the  velocitv  ratio,  and   is 

m 

designated  by  r. 

If,  now,  one  of  the  gears,  c,  be  considered  as  the  fixed  link, 
and  it  is  desired  to  examine  the  action  of  the  mechanism  when 
a  is  swung  about  ac  as  center,  it  is  evident  that  a  different 
mechanism  is  obtained.  See  §8  and  §  12.  The  action  can  be 
explained  under  the  general  laws  laid  down  in  these  sections 
but  can  be  understood  more  readily  by  reference  to  Fig.  233  C- 
Such  mechanisms  are  known  as  epicyclic  gear  trains,  because 
points  in  the  one  gear  describe  epicycloidal  curves  relatively  to 
the  other  gear.  The  name  has  no  connection  with  the  form  of 
the  gear  teeth  which  may  belong  to  the  cycloidal,  involute,  or 
any  other  system. 


TOOTHED   WHEELS  OR   GEARS. 


425 


Let  it  be  supposed  that  the  three  links  can  be  rigidly  locked 
together  and  while  so  held  are  given  a  complete  turn  about  the 
axis  ac,  in  a  clockwise  direction.  Owing  to  this,  b  will  make 
one  turn  in  a  clockwise  direction  about  its  own  axis  ab.  In 
position  i  the  arrow  is  seen  to  be  horizontal,  and  to  the  left  of  ab, 
at  2  it  is  vertical  and  above  ab,  at  3  horizontal  and  to  the  right, 
at  4  vertical  and  below,  and  at  i,  when  the  turn  about  ac  has 
been  completed,  it  is  once  more  horizontal  and  to  the  left  of  ab. 


FIG.  2330. 

The  arrow  on  b  has,  therefore,  made  a  complete  turn  about  ab 
as  axis,  and  if  one  line  of  the  rigid  body  b  has  made  such  a  turn 
the  whole  body  b  has  done  so.  But  in  swinging  the  locked 
mechanism  about  ac,  the  link  c  has  been  given  a  complete  revo- 
lution in  a  clockwise  direction.  This  is  contrary  to  the  original 
assumption  that  c  be  the  fixed  link,  i.e.,  remain  at  rest.  If, 
now,  the  mechanism  be  unlocked  and  c  be  given  a  complete 
revolution  in  a  counter-clockwise  direction  while  a  is  held  sta- 
tionary, the  result  will  be  the  same  as  though  c  had  not  been 
allowed  to  move  at  all.  But  this  counter-clockwise  revolution 
of  c  will  cause  b  to-  have  a  further  clockwise  rotation  about  its 

axis  of  —  =  r  turns.     The  total  number  of  turns  which  b  makes 

m 
about  its  axis  while  a  makes  one  turn  about  ac  will,  therefore, 

equal  i  +r. 


420 


MACHINE  DESIGN. 


Had  an  idler  been  placed  between  b  and  c,  the  result  would 
have  been  to  cause  b  to  be  given  r  turns  in  a  counter-clockwise 
or  negative  direction,  when  c  was  brought  back  to  its  original 
position  and,  consequently,  b  would  make  i  —  r  revolutions  about 
its  axis  for  one  revolution  of  a  about  ac.  This  can  be  seen 
clearly  in  Fig.  233  D,  where  b  and  c  are  purposely  made  the 


FIG.  233  D. 

same  size  so  that  r  =i  and,  hence,  i—  r  =o.  In  other  words, 
in  this  special  case  the  gear  b  does  not  rotate  about  its  axis 
at  all;  its  motion,  as  can  be  seen  from  positions  i,  2,  3,  and  4, 
being  merely  translation,  as  the  arrow  on  b  remains  always  parallel 
to  its  original  position. 

A  second  intermediate  gear,  or  idler,  would  again  reverse 
the  direction  of  £'s  motion,  making  the  revolutions  of  b  =  i+r. 

The  general  law  may  be  stated  as  follows:  — "The  number  of 
revolutions  made  by  the  last  wheel  of  an  epicyclic  train  for  each 
revolution  of  the  arm  is  equal  to  the  one  plus  the  velocity  ratio 


TOOTHED   WHEELS  OR   GEARS.  4*7 

of  the  train  if  the  number  of  axes  in  the  train  be  even,  and  one 
minus  the  velocity  ratio  of  the  train  if  the  number  of  the  axes  be 
odd.  In  the  former  case  the  wheel  turns  in  the  same  sense  as 
the  arm ;  in  the  latter  in  the  opposite  sense,  unless  the  ratio  r 
is  less  than  unity."  (Kennedy — Mechanics  of  Machinery.) 

The  same  holds  if  there  are  no  annular  gears  in  the  train  o 
if  there  are  an  even  number  of  them.     If,  however,  there  be  one 
or  any  other  odd  number  of  annular  gears  in  the  train,  the  effect 
will  be  to  transpose  the  plus  and  minus  as  well  as  the  sense  of 
rotation. 

If  the  first  wheel  of  any  epicyclic  train  has  its  axis  fixed,  but 
has  itself  a  motion  of  rotation  about  this  axis  so  that,  for  example, 
it  makes  k  revolutions  for  each  revolution  of  the  arm,  then  the 
last  wheel  of  the  train  will  make  i±r±kr  revolutions  instead 
of  i  ±r.  The  sign  of  r  is  determined  as  before  but  the  sign  of 
kr  is  plus,  if  the  rotation  of  the  first  wheel  causes  the  last  wheel 
to  rotate  in  the  same  sense  as  the  arm,  and  minus,  if  the  rotation 
of  the  first  wheel  causes  the  last  wheel  to  rotate  in  a  sense  opposite 
that  of  the  arm. 

The  only  case  which  requires  special  attention  for  fear  of 


FIG.  233  E. 

incorrectly  determining  the  number  of  axes  is  where  the  gear 
train  of  Fig.  233,  which  has  three  axes,  is  given  the  reverted  form 
shown  in  Fig.  233  E,  which  apparently  has  but  two  axes.  For 
proper  analysis  it  is  necessary  to  consider  the  reverted  train 
the  same  as  the  original  form,  i.e.,  a  double  axis  is  counted  as 
two  single  ones  in  computing  the  number  of  axes  in  the  train. 


428  MACHINE  DESIGN. 

Problem. — Find  the  number  of  revolutions  c  will  make  about 
its  axis  for  each  revolution  of  the  arm  a\  d  being  considered  as 
the  fixed  link. 

d  has  10 1  teeth  and  meshes  with  b  which  has  100  teeth,  b'  is 
keyed  to  same  shaft  as  b,  has  99  teeth,  and  meshes  with  cy  which 
has  100  teeth.  If  this  were  an  ordinary  reverted  gear  train  with 
a  as  fixed  link,  then,  remembering  that  the  angular  velocity  of 
the  last  follower  is  to  the  angular  velocity  of  the  first  driver  as 
the  product  of  the  number  of  teeth  of  the  drivers  is  to  the  product 
of  the  number  of  teeth  of  the  followers,  for  one  turn  of  d,  c  would 

make ",  =   9999  |-urns  ^  the  same  sense.    This  is  r,  the 

100X100      10000 

velocity  ratio  of  the  train.  Considering  the  train  as  an  epicyc- 
lic  one  with  d  as  fixed  link,  there  are  three  axes  and  no  annular 
gears  and  the  rule  would  be  that  for  one  turn  of  a  in  a  clock- 
wise direction  c  would  make  i  —  r  turns  about  its  axis  in 


the  same  sense,  equal  to  i -S22.  = 


i  oooo      10000 


CHAPTER  XVIII. 

SPRINGS. 

231.  Springs  Defined. — Usually  machine  members  are  required 
to  sustain  the  applied  forces  without  appreciable  yielding  and  are 
designed  accordingly;    but  certain  machine  members  are  useful 
because  of  considerable  yielding.    They  are  generally  called  springs. 

232.  Illustrations. — (a)  The  spring  of  a  safety-valve  on  a  steam- 
boiler  holds  the  valve  down  until  the  steam-pressure  reaches  the 
maximum  allowable  value;    then  it  yields  and  allows  steam  to 
escape  until  the  pressure  is  reduced,  when  it  closes  the  valve. 

(b)  The  springs  upon  which  a  locomotive-engine  is  supported 
prevent  the  transmission  of  the  full  effect  of  the  shocks,  due  to 
running,  to  the  working  parts  of  the  engine,  thereby  reducing  the 
resulting   stresses.      Car-springs    in    a    similar   manner    protect 
passengers  and  freight. 

(c)  "Bumper"  springs  reduce  stresses  in  cars  and  their  con- 
tents due  to  axial  shocks. 

(d)  The  springs  in  certain  steam-engine  governors  yield  under 
the  increased  centrifugal  force  of  the  governor  weights,  due  to 
increased  rotative  speeds,  and  allow  the  adjustment  of  the  valve- 
gear  to  the  changes  of  effort  and  load. 

(e)  Heavy  reciprocating  parts  are  often  brought  to  rest  without 
shock  and  are  then  helped  to  start  on  their  return  travel  by  the 
expanding  spring. 

(/)  A  power-hammer  strikes  a  " cushioned  blow"  because  of  the 
action  of  a  spring.  This  spring  may  be  of  steel,  rubber,  or  steam. 

(g)  Belt  connections  are  really  yielding  members  and  tend 
to  reduce  shocks  transmitted  through  them;  while  gears  (except 
rawhide  or  "  hard-fiber  ")  yield  almost  imperceptibly  and  trans- 
mit shocks  almost  unchanged. 

429 


430 


MACHINE  DESIGN. 


(h)  Long  bolts  may  become  springs  for  the  reduction  of 
stress  due  to  shock. 

(i)  Springs  may  serve  for  the  storing  of  energy  which  is 
given  out  slowly  to  actuate  light-running  mechanisms,  like  clocks. 

233.  Cantilever  Springs. — Many  springs  are  simple  cantilevers 
with  end  loads.  (See  Fig.  234.) 


.  --------  1  —  .__ 


f 


FIG.  234. 


FIG.  235. 


The  rectangular  spring  of  constant  width  b  and  height  (or 
thickness)  h,  with  a  load  F  applied  at  a  distance  /  from  the  sup- 
port gives,  from  the  laws  of  beams, 

fjbh* 


61  Ebh? 

ft  is  the  unit  stress  in  the  outer  fiber  in  pounds  per  square  inch, 
all  forces  being  expressed  in  pounds  and  all  dimensions  in  inches; 
4  is  the  total  deflection  in  inches  due  to  the  application  of  F\ 
E  is  the  modulus  of  elasticity  of  the  material  used. 

FA    fb2V 

The  work  done  or  energy  stored  =  —  =  —  —  ,  where  F=  volume, 

2 


bhl,  in  cubic  inches. 

For  a  flat  spring  of  uniform  breadth  b,  rectangular  cross- 
section,  top  surface  flat  and  lower  surface  a  parabola  in  outline, 
such  as  is  shown  in  Fig.  235, 

,2 

and    A  = 


61 


EW 


Work  done  or  energy  stored = — -. 


SPRINGS. 


431 


The  same  equations  hold  approximately  for  the  cantilever 
spring  shown  in  Fig.  236.  They  also  hold  for  the  triangular 
spring  of  constant  depth  h  shown  in  Fig.  237. 


*  —  -^  —  * 

i 

f\ 

FIG.  236. 


FIG.  237. 


In  all  of  these  cases  obviously  the  yielding  varies  inversely 
as  h3,  and  the  strength  directly  as  h2',  hence,  if  h  be  increased 
to  obtain  required  strength,  the  yielding  will  be  decreased  as  the 
cube  of  h  while  the  strength  is  increased  only  as  the  square  of  h. 
Much  of  the  requisite  yielding  is  therefore  sacrificed  if  the  strength 
is  obtained  by  increasing  h. 

Inspection  of  the  same  equations  shows  that  increasing  the 
breadth  b  to  obtain  the  required  strength  decreases  the  deflection 
in  the  same  proportion.  In  springs, 
therefore,  where  yielding  is  to  be  kept 
large,  it  is  better  to  gain  requisite 
strength  by  varying  b\  while  in  a  beam 
h  should  be  as  large  as  possible  because 
here  deflection  is  to  be  reduced  to  the 
smallest  value.  If  the  spring  is  to  be  of 
tool  steel,  hardened  and  tempered,  thin 
material  is  better  suited  to  the  operation 
of  hardening. 

As    b  is    increased  with    a    constant 

small  value  of  h,  it  may  become  too  great  for  the  available 
space.  This  difficulty  is  overcome  as  shown  by  reference  to 
Fig.  238. 

Suppose  ABC  is  a  triangular  spring  designed  for  certain  con- 


FIG.  238. 


432 


MACHINE  DESIGN. 


ditions  of  load  and  yielding.  AB  is  an  inconvenient  width. 
Divide  AB  into  equal  parts,  say  six.  Conceive  the  portion 
GFHBl  cut  off  and  placed  in  the  position  B^LK^E^  and  simi- 
larly conceive  that  EDKAi  occupies  the  position  A^MKiEi, 
and  the  two  parts  are  rigidly  joined  along  the  line  K^^.  Also 
conceive  the  portion  ADE  moved  to  AiDiEi,  and  BFG  moved 
to  BiDiEi,  and  that  they  are  rigidly  joined  along  the  line  D\E\. 

The  amount  of  material  is  unchanged.  The  bending  force  is 
applied  in  nearly  the  same  way  to  the  portions  whose  position  is 
changed.  The  leaf  spring  is  therefore  practically  equivalent 
to  the  triangular  spring  from  which  it  is  made. 

The .  following  equations  are  given  by  Prof.  J.  B.  Peddle,* 
for  leaf  springs  having  both  full  and  pointed  leaves  of  equal  base 
number  of  full  length  leaves 


width.     Let  r 


total  number  of  leaves 


FIG.  2384. 
For  semi-elliptic  spring,  Fig.  2 38 A: 


FIG.  2385. 


F  = 


fbh? 
3*"' 


b  =  number  of  leaves X  width  per  leaf,  inches; 

h= thickness  per  leaf,  inches; 

/=unit  stress  in  outer  fiber,  pounds  per  square  inch. 

*  American  Machinist,  April  17,  1913.     Prepared  for  Halsey's  "Handbook  for 
Machine  Designers." 


SPRINGS.  433 

2l2fK 

A  =    ,  _  . 


For  full  elliptic  spring,  Fig.  23 85. 

fbk* 


hE 


234.  Springs  for  Axial  Loads. — Many  springs  are  subjected 
to  axial  loads;  they  are  usually  helical  in  form  as  shown  in 
Fig.  239. 


FIG.  239. 

F  may  act  to  stretch  or  compress  the  spring. 
Consider  the  cross-section  of  the  rod  to  be  circular. 
Let  F  =  load  in  pounds  ; 

d=  diameter  of  rod  in  inches; 
a  =  mean  radius  of  coil  in  inches; 
N  =  number  of  coils; 

/=  developed  length  of  spring  in  inches  =  27ra7V; 
fs=  allowable  unit  shearing  stress  in  outer  fiber  in  pounds 

per  square  inch  ; 

E8  =  modulus  of  shearing  elasticity  =f  E; 
J  =  extension  or  compression  in  inches. 
7?hen  the  following  equations  may  be  developed: 


i6Fa  s 

and    ^^'      Work  done 


434  MACHINE  DESIGN. 

In  a  helical  spring  for  an  axial  load  using  a  rectangular  cross- 
section  of  wire  the  axial  height  of  wire  =  h  and  radial  breadth  =  b, 
the  equations  become,  approximately, 


It  will  take  one  and  a  half  times  as  much  material  to  make  a  spring 
of  this  type  as  it  would  to  make  a  round-wire  helical  spring  of 
equal  strength. 

235.  Springs  for  Torsional  Movements. — Many  springs  come 


FIG.  240. 

under  class  (i)  as  mentioned  above.  The  general  case  is  shown 
in  Fig.  240.  The  spring  is  a  spiral  of  flat  wire  with  an  axial 
height  b  and  radial  depth  h. 

F=  turning  force  in  pounds; 

a  =  lever-arm  in  inches; 

/6  =  unit  stress  in  outer  fiber,  pounds  per  square  inch; 

/=  developed  length  of  spring; 

#  =  angular  deflection  ; 

J  =  distance  inches  moved  through  by  the  point  of  appli- 
cation of  F. 

fbbh2 
Then  F=  --  •     and     A  =  a&  = 


. 
6a  Ebh? 

fb2V 

Work  done  or  energy  stored  =  —  —  . 

6E 

236.  Materials  and  Allowable    Stresses.  —  Springs    may,    of 
course,  be  made  of  any  material  having  elastic  strength;    buJ 


SPRINGS.  43$ 

spring  steel  is  the  material  most  frequently  used.  In  its  un- 
tempered  or  soft  condition  it  has  an  elastic  limit  of  from  70,000 
to  90,000  Ibs.  per  square  inch.  Tempered,  or  hard  drawn,  its 
elastic  limit  may  be  from  110,000  to  140,000  Ibs.  per  square 
inch. 

A  unit  bending  stress  of  80,000  Ibs.  and  a  shearing  stress  of 
60,000  may  be  used  with  safety. 

£  =  30,000,000     and    £,=  12,500,000. 

For  round  rod  torsional  springs  the  following  values  may 
be  used. 

Diam.  of  wire,  in.  Safe  unit  stress,  /*. 

1  70,000 

§  6o,OOO 

i  50,000 

For  further  information  the  reader  is  referred  to  Reuleaux's 
«  Constructor,"  Trans.  A.  S.  M.  E.,  Vols.  V  and  XVI,  and  Hal- 
sey's  '*  Handbook  for  Machine  Designers." 


CHAPTER   XIX. 

MACHINE  SUPPORTS. 

237.  General  Laws  for  Machine    Supports. — The  single-box 
pillar  support  is  best  and  simplest  for  machines  whose  size  and 
form  admit  of  its  use.     When  a  support  is  a  single  continuous 
member,  its  design  should  be  governed  by  the  following  principles : 

I.  The  amount  of  material  in  the  cross-section  is  determined 
by  the  intensity  of  the  load.      If  vibrations  are  also  to  be  sus- 
tained, the  amount  of  material  must  be  increased  for  this  purpose. 

II.  The  vertical  center  line  of  the  support  should  coincide 
with  the  vertical  line  through  the  center  of  gravity  of  the  part 
supported. 

III.  The  vertical  outlines  of  the  support  should  taper  slightly 
and  uniformly  on  all  sides.     If  they  were  parallel  they  would 
appear  nearer  together  at  the  bottom. 

IV.  The  external  dimensions  of  the  support  must  be  such 
that  the  machine  has  the  appearance  of  being  in  stable  equilibrium. 
The  outline  of  all  heavy  members  of  the  machine  supported  must 
be  either  carried  without  break  to  the  foundation,   or  if  they 
overhang,  must  be  joined  to  the  support  by  means  of  parabolic 
outlines,  or  by  the  straight  lines  of  the  brace  form. 

238.  Illustration. — In  Fig.  241  the  first  three  principles  may 
be   fulfilled,    but   there  is   an   appearance   of   instability.     It  is 
because  the  outline  of  the  "housing"  overhangs.     It  should  be 
carried  to  the  foundation  without  break  in  the  continuity  of  the 

metal,  as  in  Fig.  242. 

436 


MACHINE  SUPPORTS. 


437 


239.  Divided  Supports. — When  the  support  is  divided  up  into 
several  parts,  modification  of  these  principles  becomes  necessary, 
as  the  divisions  require  separate  treatment.  This  question  may 


FIG.  241.  FIG.  242. 

be  illustrated  by  lathe  supports.  In  Fig.  243  are  shown  three 
forms  of  support  for  a  lathe,  seen  from  the  end.  For  stability 
the  base  needs  to  be  broader  than  the  bed.  In  A  the  width  of 
base  necessary  is  determined  and  the  outlines  are  straight  lines. 
The  unnecessary  material  is  cut  away  on  the  inside,  leaving  legs 
which  are  compression  members  of  correct  form.  The  cross - 
brace  is  left  to  check  any  tendency  to  buckle.  For  convenience 
to  the  workmen  it  is  desirable  to  narrow  this  support  somewhat 


FIG   243. 

without  narrowing  the  base.  The  cross -brace  converts  the  single 
compression  member  into  two  compression  members.  It  is  allow- 
able to  give  these  different  angles  with  the  vertical.  This  is  done 
in  B  and  the  straight  lines  are  blended  into  each  other  by  a  curve. 
C  shows  a  common  incorrect  form  of  lathe  support,  the  compres- 
sion members  from  the  cross-brace  downward  being  curved. 
There  is  no  reason  for  this  curved  form.  It  is  less  capable  of 


438 


MACHINE  DESIGN. 


bearing  its  compressive  load  than  if  it  were  straight,  and  is  no 
more  stable  than  the  form  B,  the  width  of  base  being  the  same. 


I    o 


FIG.  244. 

Consider  the  lathe  supports  from  the  front.  Four  forms  are 
shown  in  Fig.  244.  If  there  were  any  force  tending  to  move  the 
bed  of  the  lathe  endwise  the  forms  B  and  C  would  be  allowable. 
But  there  is  no  force  of  this  kind,  and  the  correct  form  is  the 
one  shown  in  D.  Carrying  the  foot  out  as  in  A,  B,  and  C  in- 
creases the  distance  between  supports  (the  bed  being  a  beam 
with  end  supports  and  the  load  between) ;  this  increases  the  de- 
flection and  the  fiber  stress  due  to  the  load.  This  increase  in 
stress  is  probably  not  of  any  serious  importance,  but  the  prin- 
ciple should  be  regarded  or  the  appearance  of  the  machine  will 
not  be  right.  If  the  supports  were  joined  by  a  cross-member, 


FIG    745. 

as  in  Fig.  245,  they  would  be  virtually  converted  into  a  single 
support,  and  should  then  taper  from  all  sides. 

240.  Three-point  Support. — If  a  machine  be  supported  on 
a  single -box  pillar,  change  in  the  form  of  the  foundation  cannot 
induce  stress  in  the  machine  frame  tending  to  change  its  form. 
If,  however,  the  machine  is  supported  on  four  or  more  legs  the 


MACHINE  SUPPORTS.  439 

foundation  might  sink  away  from  one  or  more  of  them  and  leave 
a  part  unsupported.  This  might  cause  torsional  or  flexure  stress 
in  some  part  of  the  machine,  which  might  change  its  form  and 
interfere  with  the  accuracy  of  its  action. 

But  if  the  machine  is  supported  on  three  points  this  cannot 
occur,  because  if  the  foundation  should  sink  under  any  one  of 
the  supports  the  support  would  follow  and  the  machine  would 
still  rest  on  three  points.  When  it  is  possible,  therefore,  a  ma- 
chine which  cannnot  be  carried  on  a  single  pillar  should  be  sup- 
ported on  three  points.  Many  machines  are  too  large  for  three- 
point  support,  and  the  resource  is  to  make  the  bed,  or  part  sup- 
ported, of  box  section  and  so  rigid  that  even  if  some  of  the  legs 
should  be  left  without  foundation  the  part  supported  would  still 
maintain  its  form.  More  supports  are  often  used  than  are  necessary. 
Thus,  if  a  lathe  has  two  pairs  of  legs  like  those  shown  in  B, 
Fig.  243,  and  these  are  bolted  firmly  to  the  bed,  there  will  be 
four  points  of  support.  But  if,  as  suggested  by  Professor  Sweet, 
one  of  these  pairs  be  connected  to  the  bed  by  a  pin  so  that  the 
support  and  the  bed  are  free  to  move  relatively  to  each  other 
about  the  pin,  as  in  Fig.  246,  then  this  is  equivalent  to  a  single 
support,  and  the  bed  will  have  three  points  of  support,  and  will 
maintain  its  form  independently  of  any  change  in  the  foundation. 
This  is  of  special  importance  when  the  machines  are  to  be  placed 
upon  yielding  floors. 

241.  Reducing  Number  of  Supports. — Fig.  247  shows  another 


FIG.  246.  FIG.  247. 

case  in  which  the  number  of  supports  may  be  reduced  without 
sacrifice.     In  A  three  pairs  of  legs  are  used.     There  are  therefore 


440 


MACHINE  DESIGN. 


six  points  of  support.  In  B  two  pairs  of  legs  are  used  and  one 
may  be  connected  by  a  pin,  and  there  will  be  but  three  points  of 
support.  The  chance  of  the  bed  being  strained  from  changing 
foundation  has  been  reduced  from  6  in  A  to  o  in  B.  The  total 
length  of  bed  is  12  feet,  and  the  unsupported  length  is  6  feet 
in  both  cases. 

242.  Further   Correct   Methods. — Figs.    248   and    249    show 
correct  methods  of  support  for  small  lathes  and  planers,  due  to 


Jl 


FIG.  248. 


FIG.  249. 


Professor  Sweet.  In  Fig.  248  the  lathe  "head-stock  "  has  its 
outlines  carried  to  the  foundation  by  the  box  pillar;  a  represents 
a  pair  of  legs  connected  to  the  bed  by  a  pin  connection,  and 
instead  of  being  placed  at  the  end  of  the  bed  it  is  moved  in  some- 
what, the  end  of  the  bed  being  carried  down  to  the  support  by  a 
para,bolic  outline.  The  unsupported  length  of  bed  is  thereby 
decreased,  the  stress  on  the  bed  is  less,  and  the  bed  will  maintain 
its  form  regardless  of  any  yielding  of  the  floor  or  foundation. 
In  Fig.  249  the  housings,  instead  of  resting  on  the  bed  as  is  usual 
in  small  planers,  are  carried  to  the  foundation,  forming  two  of 
the  supports;  the  other  is  at  a  and  has  a  pin  connection  with  the 
bed,  which  being  thus  supported  on  three  points  cannot  be  twisted 
or  flexed  by  a  yielding  foundation. 


CHAPTER  XX. 

MACHINE    FRAMES. 

243.  Open-side  Frame. — Fig.  250  shows  an  open -side  frame, 
such  as  is  used  for  punching  and  shearing  machines.  During  the 
action  of  the  punch  or  shear  a  force  is  applied  to  the  frame  tending 
to  separate  the  jaws.  This  force  may  be  represented  in  magnitude, 
direction,  and  line  of  action  by  P.  It  is  required  to  find  the 
resulting  stresses  in  the  three  sections  AB,  CD,  and  EF.  Con- 
sider AB.  Let  the  portion  above  this  section  be  taken  as  a  -free 
body.  The  force  P,  Fig.  251,  and  the  opposing  resistances  to 


FIG.  250. 

deformation  of  the  material  at  the  section  AB,  are  in  equilibrium. 
Let  H  be  the  projection  of  the  gravity  axis  of  the  section  AB, 
perpendicular  to  the  paper.  Two  equal  and  opposite  forces,  P\ 
and  P2,  rnay  be  applied  at  H  without  disturbing  the  equilibrium. 
Let  PI  and  P2  be  each  equal  to  P,  and  let  their  line  of  action  be 
parallel  to  that  of  P.  The  free  body  is  now  subjected  to  the 
action  of  an  external  couple,  PI,  and  an  external  force,  P2.  The 
couple  produces  flexure  about  H,  and  the  force  P2  produces  tensile 
stress  in  the  section  AB.  The  flexure  results  in  a  tensile  stress 

varying  from  a  maximum  value  in  the  outer  fiber  at  A  to  zero 

441 


442 


MACHINE  DESIGN. 


at  H,  and  a  compressive  stress  varying  from  a  maximum  in  the 
outer  fiber  at  B  to  zero  at  H.  This  may  be  shown  graphically 
at  JK.  The  ordinates  of  the  line  LM  represent  the  varying 
stress  due  to  flexure;  while  ordinates  between  LM  and  NO 
represent  the  uniform  tensile  stress.  This  latter  diminishes  the 
compressive  stress  at  B,  and  increases  the  tensile  stress  at  A. 

The  tensile  stress  per  square 
inch  at  A  therefore  equals  /+/i  ; 
where  /  equals  the  unit  fiber 
stress  due  to  flexure  at  A,  and/! 
equals  the  unit  tensile  stress  due 


FIG.  252. 


FIG.  251. 


in  which  c  =  the  distance   from 
the  gravity  axis  to  the  outer  fiber 


=  AH,  and  7  =  the  moment  of  inertia  of  the  section  about  H,  and 
A  =  area  of  the  cross-section  AB. 

Were  the  vertical  bounding  walls  at  A  and  B  curved  surfaces, 
the  locus  of  neutral  axes  (centroid)  through  H  would  be  a  curve. 
In  this  case  according  to  E.  S.  Andrews:  * 

At  inner  edge  of  section,    /&  =  P[i  +(Klci +  &*)]+ A. 
At  outer  edge,  fc=P[(Clc2+k2) -i]+A. 

Ci  =  distance  from  centroid  to  inner  edge,  inches; 
C2=  distance  from  centroid  to  outer  edge,  inches; 

k  =  radius  of  gyration  of  section  about  centroid  in  inch  units; 
A  =  cross-sectional  area,  square  inches ; 
ff=  (7.84* +0.42)*  (8* -3); 
C=  (5-0.16) -s- (5 +0.2); 

s  =  radius  of  curvature  of  centroid  divided  by  (£1+^2). 

244.  Problem. — Let  it  be  required  to  design  the  frame  of  a 
machine  to  punch  j-inch  holes  in  ^-inch  steel  plates,  18  inches 

*  American  Machinist,  Sept.  5,  1912. 


MACHINE  FRAMES.  443 

from  the  edge.  The  surface  resisting  the  shearing  action  of  the 
punch  =  7:Xi"Xi"  =  i.i8  square  inch.  The  ultimate  shearing 
strength  of  the  material  is,  say,  50,000  Ibs.  per  square  inch.  The 
total  force  P,  which  must  be  resisted  by  the  punch  frame  =  50,000 
X  1.  18  =  59,000  Ibs. 

The  material  and  form  for  the  frame  must  first  be  selected. 
The  form  is  such  that  forged  material  is  excluded,  and  difficulties 
of  casting  and  high  cost  exclude  steel  casting.  The  material, 
therefore,  must  be  cast  iron.  Often  the  same  pattern  is  used 
both  for  the  frame  of  a  punch  and  shear.  In  the  latter  case, 
when  the  shear  blade  begins  and  ends  its  cut,  the  force  is  not 
applied  in  the  middle  plane  of  the  frame,  but  considerably  to  one 
side,  and  a  torsional  stress  results  in  the  frame.  Combined  torsion 
and  flexure  are  best  resisted  by  members  of  box  form.  The  frame 
will  therefore  be  made  of  cast  iron  and  of  box  section.  The  dimen- 
sion AB  may  be  assumed  so  that  its  proportion  to  the  "reach" 
of  the  punch  appears  right;  the  width  and  thickness  of  the  cross- 
section  ir  ay  also  be  assumed.  From  these  data  the  maximum 
stress  in  the  outer  fiber  may  be  determined.  If  this  is  a  safe 
value  for  the  material  used  the  design  will  be  right. 

Let  the  assumed  dimensions  be  as  shown  in  Fig.  252.     Then 

A  =Mi  -  W2  =  78  square  inches. 


12 

=  3002  bi-quadratic  inches. 

c=di-^-2=9";  /=the  reach  of  the  punch  +c  =  27";  P  =59,000  Ibs., 
as  determined  above.     Then 

P 


Pic    59000X27X9 

=4776> 


/i  +/  =  5532  =maximum  stress  in  the  section. 


44** 


MACHINE  DESIGN. 


The  average  strength  of  cast  iron  such  as  is  used  for  machinery 
castings,  is  about  20,000  Ibs.  per  square  inch.  The  factor  of 
safety  in  the  case  assumed  equals  20,000-^5532=3.65.  This  is 
too  small.  There  are  two  reasons  why  a  large  factor  of  safety 
should  be  used  in  this  design:  I.  When  the  punch  goes  through 
the  plate  the  yielding  is  sudden  and  a  severe  stress  results.  This 
stress  has  to  be  sustained  by  the  frame,  which  for  other  reasons 
is  made  of  unresilient  material.  II.  Since  the  frame  is  of  cast 
iron,  there  will  necessarily  be  shrinkage  stresses  which  the  frame 
must  sustain  in  addition  to  the  stress  due  to  external  force's. 
These  shrinkage  stresses  cannot  be  calculated  and  therefore  can 
only  be  provided  against  by  a  large  factor  of  safety. 

Cast  iron  is  strong  to  resist  compression  and  weak  to  resist 
tension,  and  the  maximum  fiber  stress  is  tension  on  the  inner 
side.  The  metal  can  therefore  be  more  satisfactorily  distributed 
than  in  the  assumed  section,  by  being  thickened  where  it  sustains 
tension,  as  at  A ,  Fig.  253.  If,  how- 
ever, there  is  a  very  thick  body 
of  metal  at  a,  sponginess  and  ex- 
cessive shrinkage  would  result. 
The  form  B  would  be  better,  the 
metal  being  arranged  for  proper 
cooling  and  for  the  resisting  of 
flexure  stress. 


FIG..  253. 


FIG.  254. 


Dimensions  may  be  assigned  to  a  section  like  B  and  the 
cross-section  may  be  checked  for  strength  as  before.  See  Fig.  254. 
GG,  a  line  through  the  center  of  gravity  of  the  section,  is  found  to 
be  at  a  distance  of  7.05  inches  from  the  tension  side.*  The 


*  A  simple  and  satisfactory  method  for  obtaining  a  close  approximation  to 


MACHINE  FRAMES.  445 

required  values  are  as  follows:  £  =  7.05  inches;  /  =  reach  of 
punch  +£  =  18  +  7.05  =25.05  inches;  .4=156.25  square  inches; 
7  =  5032.5  bi-quadratic  inches;  P  =  59,000  Ibs. 


^59000x25.05x7.05  - 

I  5032-5 

/i+/  =  2448  Ibs.  =  maximum  fiber  stress  in  the  section.  The 
factor  of  safety  =  20,000^2448=8.17.*  This  section,  therefore, 
fulfills  the  requirement  for  strength,  and  the  material  is  well 
arranged  for  cooling  with  little  shrinkage,  and  without  spongy 
spots.  The  gravity  axis  may  be  located,  and  the  value  of  /  deter- 
mined by  graphic  methods.  See  Hoskins's  "Graphic  Statics."  | 
Let  the  section  CD,  Fig.  250,  be  considered.  Fig.  255  shows 
the  part  at  the  left  of  CD  free.  K  is  the  projection  of  the  gravity 
axis  of  the  section.  As  before,  put  in  two  opposite  forces,  PS 
and  P4,  equal  to  each  other  and  to  P,  and  having  their  common 
line  of  action  parallel  to  that  of  P,  at  a  distance  l\  from  it.  P 
and  P4now  form  a  couple,  whose  moment  =P/i,  tending  to  produce 
flexure  about  K.  P3  must  be  resolved  into  two  components, 
one  PS/,  at  right  angles  to  the  section  considered,  tending  to 
produce  tensile  stress;  and  the  other  JK,  parallel  to  the  section, 
tending  to  produce  shearing  stress.  The  greatest  unit  tensile 

the  true  gravity  axis  of  irregular  figures  is  as  follows:  On  a  piece  of  thin  but  uni- 
form cardboard  lay  out  the  figure  to  scale.  Cut  it  out  carefully  with  a  sharp 
knife.  Balance  the  figure  exactly,  by  trial,  on  a  knife-edge.  The  line  of  contac! 
with  the  knife-edge  is  the  gravity  axis.  Its  position  may  be  marked  and  its  loca- 
tion measured  to  scale. 

*  This  discussion  neglects  the  action  of  gravity  which  would  exert  a  counter- 
balancing moment,  reducing  the  maximum  tensile  fiber  stress  below  the  value 
found.  This  makes  the  actual  factor  of  safety  greater  than  the  apparent  iactor 
of  safety. 

f  The  student  will  be  familiar  with  analytical  methods  for  their  determina 
tion  from  his  study  of  the  "Mechanics  of  Materials." 


446 


MACHINE  DESIGN. 


stress  in  this  section  will  equal  the  sum  of  that  due  to  flexure 
and  that  due  to  tension 


T  77- 

The  greatest  unit  shear  =/«=~r- 

In  the  section  FE,  Fig.  250,  which  is  parallel  to  the  line  of 
action  of  P,  equal  and  opposite  forces,  each  =  P,  may  be  intro- 
duced, as  P$  and  P&.  P  and  P§  will  then  form  a  couple  with  an 


FIG.  256. 

arm  /2,  and  P5  will  be  wholly  applied  to  produce  shearing  stress. 
The  maximum  unit  tensile  stress  in  this  section  will  be  that  due 
to  flexure,  ]  =  Pl^c -.- 1 ,  and  the  maximum  unit  shear  will  be 
j8  =  P+A.  Any  section  may  be  thus  checked. 


MACHINE  FRAMES. 


447 


The  dimensions  of  several  sections  being  found,  the  outline 
curve  bounding  them  should  be  drawn  carefully,  to  give  good 
appearance.  The  necessary  modifications  of  the  frame  to  pro- 
vide for  support,  and  for  the  constrainment  of  the  actuating 
mechanism,  may  be  worked  out  as  in  Fig.  256.  A  is  the  pLuoii 
on  the  pulley  shaft  from  which  the  power  is  received;  B  is  the 
gear  on  the  main  shaft ;  C,  D,  and  G  are  parts  of  the  frame  added 
to  supply  bearings  for  the  shafts;  E  furnishes  the  guiding  surfaces 
for  the  punch  "slide."  The  method  of  supporting  the  frame  is 
shown,  the  support  being  cut  under  at  F  for  convenience  to  the 
workman.  The  parts  C,  D,  E,  and  G  can  only  be  located  after 
the  mechanism  train  has  been  designed. 


FIG.  257. 

245.  Slotting-machine  Frame. — See  Fig.  257.  It  is  specified 
that  the  slotter  shall  cut  at  a  certain  distance  from  the  edge  of 
any  piece,  and  the  dimension  AH  is  thus  determined.  The 
table  G  must  be  held  at  a  convenient  height  above  the  floor,  and 
RK  must  provide  for  the  required  range  of  "feed."  K  is  cut 
under  for  convenience  of  the  workman,  and  carried  to  the  floor 
line  as  shown.  It  is  required  to  "slot  "  a  piece  of  given  vertical 
dimension,  and  the  distance  from  the  surface  of  the  table  to  E 
is  thus  determined.  Let  the  dimension  LM  be  assumed  so  that 


448  MACHINE  DESIGN. 

it  shall  be  in  proper  proportion  to  the  necessary  length  and  height 
of  the  machine.  The  curves  LS  and  M T  may  be  drawn  for 
bounding  lines  of  a  box  frame  to  support  the  mechanism.  M 
should  be  carried  to  the  floor  line  as  shown,  and  not  cut  under. 
None  of  the  part  DNE,  nor  that  which  serves  to  support  the 
cone  and  gears  on  the  other  side  of  the  frame,  should  be  made 
flush  with  the  surface  LSTM,  because  nothing  should  interfere 
with  the  continuity  of  the  curves  LS  and  TM.  The  supporting 
jrame  0}  a  machine  should  be  clearly  outlined,  and  other  parts  should 
appear  as  attachments.  The  member  VW  should  be  designed  so 
that  its  inner  outline  is  nearly  parallel  to  the  outline  of  the  cone 
pulley,  and  should  be  joined  to  the  main  frame  by  a  curve.  The 
outer  outline  should  be  such  that  the  width  of  the  member  increases 
slightly  from  W  to  F,  and  should  also  be  joined  to  the  main 
frame  by  a  curved  outline.  In  any  cross-section  of  the  frame, 
as  XX,  the  amount  of  metal  and  its  arrangement  may  be  con- 
trolled by  the  core.  It  is  dictated  by  the  maximum  force,  P, 
which  the  tool  can  be  required  to  sustain.  The  tool  is  carried  by 
the  slider  of  a  slider-crank  chain.  Its  velocity  varies,  therefore, 
from  a  maximum  near  mid-stroke,  to  zero  at  the  upper  and  lower 
ends  of  its  stroke.  The  belt  which  actuates  the  mechanism  runs 
on  one  side  of  the  steps  of  the  cone  pulley,  at  a  constant  velocity. 
Suppose  that  the  tool  is  set  (accidentally)  so  that  it  strikes  the 
table  just  before  the  slider  has  reached  the  lower  end  of  its  stroke. 
The  resistance,  R,  offered  by  the  tool  to  being  stopped,  multiplied 
by  its  (very  small)  velocity,  equals  the  difference  of  belt  tension 
multiplied  by  the  belt  velocity  (friction  and  inertia  neglected).* 
R,  therefore,  would  vary  inversely  as  the  slider  velocity,  and  hence 
may  be  very  great.  Its  maximum  value  is  indeterminate.  A 
"breaking  piece  "  may  be  put  in  between  the  tool  and  the  crank. 
Then  when  R  reaches  a  certain  value,  the  breaking  piece  fails. 

*  See  Chapter  V. 


MACHINE  FRAMES.  449 

The  stress  in  the  stress-members  of  the  machine  is  thereby  limited 
to  a  certain  definite  value.  From  this  value  the  frame  may  be 
designed.  Let  P  =  up  ward  force  against  the  tool  when  the  break- 
ing piece  fails.*  Let  /  =  the  horizontal  distance  from  the  line  of 
action  P  to  the  gravity  axis  of  the  section  XX.  Then  the  section 
XX  sustains  flexure  stress  caused  by  the  moment  P/,  and  tensile 
stress  equal  to  P.  The  maximum  unit  stress  in  the  section 

Pic      P 


A  section  may  be  assumed  and  checked  for  safety,  as  for  the 
punching-machine  in  §  244. 

246.  Stresses  in  the  Frame  of  a  Side-crank  Steam-engine. — • 
Fig.  258  is  a  sketch  in  plan  of  a  side -crank  engine  of  the  "  girder 


bed ''  type.  The  supports  are  under  the  cylinder  C,  the  main 
bearing  £,  and  the  out-board  bearing  D.  A  force  P  is  applied 
in  the  center  line  of  the  cylinder,  and  acts  alternately  toward  the 
right  and  toward  the  left.  In  the  first  case  it  tends  to  separate 
the  cylinder  and  main  shaft;  and  in  the  second  case  it  tends  to 
bring  them  nearer  together.  The  frame  resists  these  tendencies 
with  resulting  internal  stresses. 

Let  the  stresses  in  the  section  AB  be  considered.  The  end  of 
ths  frame  is  shown  enlarged  in  Fig.  259.  If  the  pressure  from  the 
piston  is  toward  the  right,  the  stresses  in  AB  will  be:  I.  Flexure 

*  P  is  limited  to  the  friction  due  to  screwing  up  the  four  bolts  which  hold  the 
tool. 


45° 


MACHINE  DESIGN. 


due  to  the  moment  P/,  resulting  in  tensile  stress  below  the  gravi'ty 
axis,  TV,  with  a  maximum  value  at  b,  and  a  compressive  stress 
above  N  with  a  maximum  value  at  a.  II.  A  direct  tensile  stress, 
=  P,  distributed  over  the  entire  section,  resulting  in  a  unit  stress  = 
P  +  A  =}i  Ibs.  per  square  inch.  This  is  shown  graphically  at  n, 
Fig.  259.  a\bi  is  a  datum  line  whose  length  equals  ab.  Tensions 
are  laid  off  toward  the  right  and  compressions  toward  the  left. 

Jheiai 


(«) 


FlG.  259. 

The  stress  due  to  flexure  varies  directly  as  the  distance  from  the 
neutral  axis  JVi,  being  zero  at  NI.  If,  therefore,  b^i  represents 
the  tensile  stress  in  the  outer  fiber,  then  c\k\  drawn  through  N\ 
will  be  the  locus  of  the  ends  of  horizontal  lines,  drawn  through 
all  points  of  a\bi,  representing  the  intensity  of  stress  in  all  parts 
of  the  section,  due  to  flexure.  If  c\d\  represent  the  unit  stress 
due  to  direct  tension,  then,  since  this  is  the  same  in  all  parts  of 
the  section,  it  will  be  represented  by  the  horizontal  distance  be- 
tween the  parallel  lines  c\k\  and  A\e\.  This  uniform  tension 
increases  the  tension  bic\  due  to  flexure,  causing  it  to  become  b±d± ; 
and  reduces  the  compression  kidit  causing  it  to  beccme  e\a\. 
The  maximum  stress  in  the  section  is  therefore  tensile  stress  in 
the  lower  outer  fiber,  and  is  equal  to  b\d\. 

When  the  force  P  is  reversed,  acting  toward  the  left,  the 
stresses  in  the  section  are  as  shown  at  m,  Fig.  259:  compression 
due  to  flexure  in  the  lower  outer  fiber  equal  to  c2b2]  tension  due 
to  flexure  in  the  upper  outer  fiber  equal  to  a2k2;  and  uniform 
compression  over  the  entire  surface  equal  to  d2c2.  This  latter 
increases  the  compression  in  the  lower  outer  fiber  from  b2c2  to 
and  decreases  the  tension  in  the  upper  outer  fiber  from  a2k2 


MACHINE  FRAMES. 


451 


to  a2€2-  The  maximum  stress  in  the  section  is  therefore  compres- 
sion in  the  lower  outer  fiber  equal  to  b2d2.  The  maximum  stress, 
therefore,  is  always  in  the  side  of  the  frame  next  to  the  connecting- 
rod. 

If  the  gravity  axis  of  the  cross-section  be  moved  toward  the 
connecting-rod,  the  stress  in  the  upper  outer  fiber  will  be  increased, 
and  that  in  the  lower  outer  fiber  will  be  proportionately  decreased. 
The  gravity  axis  may  be  moved  toward  the  connecting-rod  by 
increasing  the  amount  of  material  in  the  lower  part  of  the  cross- 
section  and  decreasing  it  in  the  upper  part. 

The  stress  in  any  other  section  nearer  the  cylinder  will  be  due 
to  the  same  force,  P,  as  before;  but  the  moment  tending  to  pro- 
duce flexure  will  be  less,  because  the  lever  arm  of  the  moment  is 
less  and  the  force  constant. 

247.  Heavy-duty  Engine  Frame. — Suppose  the  engine  frame 
to  be  of  the  type  which  is  continuous  with  the  supporting  part  as 
shown  in  Fig.  260.  Let  Fig.  261  be  a  cross-section,  say  at  AB. 
O  is  the  center  of  the  cylinder.  The  force  P  is  applied  at  this 


FIG 


point  perpendicular  to  the  paper.  C  is  the  center  of  gravity  of 
the  section  (the  intersection  of  two  gravity  axes  perpendicular 
to  each  other,  found  graphically).  Join  C  and  O,  and  through 


452  MACHINE  DESIGN. 

C  draw  XX  perpendicular  to  CO.  Then  XX  is  the  gravity 
axis  about  which  flexure  will  occur.*  The  dangerous  stress  will 
be  at  F,  and  the  value  of  c  will  be  the  perpendicular  distance  from 
F  to  XX.  The  moment  of  inertia  of  the  cross-section  about 
XX  may  be  found,  =7;  /,  the  lever-arm  of  P,  =OC.  The 
stress  at  F,  j  +  fi  must  be  of  safe  value. 

Pk 

]=—j-j  in  known  terms. 

p 

fi  =  —     —2 -; — >  in  known  terms. 

area  of  section 

248.  Closed  Frames. — Fig.  262  shows  a  closed  frame.  The 
members  G  and  H  are  bolted  rigidly  to  a  cylinder  C  at  the  top,  and 
to  a  bedplate,  DD,  at  the  bottom.  A  force  P  may  act  in  the  center 
line,  either  to  separate  D  and  C,  or  to'  bring  them  nearer  together. 
The  problem  is  to  design  G,  H,  and  D  for  strength.  If  the  three 
members  were  "pin  connected"  (see  Fig.  263),  the  reactions  of 
C  upon  A  and  B  at  the  pins  would  act  in  the  lines  EF  and  GH. 
Then  if  P  acts  to  bring  D  and  C  nearer  together,  compression 
results  in  A,  the  line  of  action  being  EF;  compression  results 
in  B,  the  line  of  action  being  GH.  These  compressions  being  in 
equilibrium  with  the  force  P,  their  magnitude  may  be  found  by 
the  triangle  of  forces.  From  these  values  A  and  B  may  be  designed. 
C  is  equivalent  to  a  beam  whose  length  is  /,  supported  at  both 

*  This  is  not  strictly  true.  If  OC  is  a  diameter  of  the  "ellipse  of  inertia," 
flexure  will  occur  about  its  conjugate  diameter.  If  the  section  of  the  engine  frame 
is  symmetrical  with  respect  to  a  vertical  axis,  OC  is  vertical,  and  its  conjugate 
diameter  XX  is  horizontal.  Flexure  would  occur  about  XX,  and  the  angle  be- 
tween OC  and  XX  would  equal  90°.  As  the  section  departs  from  symmetry 
about  a  vertical,  XX,  at  right  angles  to  OC,  departs  from  OC's  conjugate,  and 
hence  does  not  represent  the  axis  about  which  flexure  occurs.  In  sections  like 
Fig.  259,  the  error  from  making  ^=90°  is  unimportant.  When  the  departure 
from  symmetry  is  very  great,  however,  OC's  conjugate  should  be  located  and 
used  as  the  axis  about  which  flexure  occurs.  For  method  of  drawing  "ellipse  of 
inertia"  see  Hoskins's  "  Graphic  Statics." 


MACHINE  FRAMES. 


453 


ends,  sustaining  a  transverse  load  P,  and  tension  equal  to  the 
horizontal  component  of  the  compression  in  A  or  B.  The  data 
for  its  design  would  therefore  be  available.  Reversing  the  direc- 
tion of  P  reverses  the  stresses;  the  compression  in  A  and  B  becomes 
•tension;  the  flexure  moment  tends  to  bend  C  convex  downward 
instead  of  upward,  and  the  tension  in  C  becomes  compression. 
But  when  the  members  are  bolted  rigidly  together,  as  in 

FIG.  262. 


FIG.  263. 

Fig.  262,  the  lines  of  the  reactions  are  indeterminate.  Assump- 
tions must  therefore  be  made.  Suppose  that  G  is  attached  to  D 
by  bolts  at  E  and  A.  Suppose  the  bolts  to  have  worked  slightly 
loose,  and  that  P  tends  to  bring  C  and  D  nearer  together.  There 
would  be  a  tendency,  if  the  frame  yields  at  all,  to  relieve  pressure 
at  E  and  to  concentrate  it  at  A.  The  line  of  the  reaction  would 
pass  through  A  and  might  be  assumed  to  be  perpendicular  to 


454  MACHINE  DESIGN. 

the  surface  AE.  Suppose  that  P  is  reversed  and  that  the  bolts 
at  A  are  loosened,  while  those  at  E  are  tight.  The  line  of  the 
reaction  would  pass  through  E,  and  might  be  assumed  to  be 
perpendicular  to  EA.  MN  is  therefore  the  assumed  line  of  the 
reaction,  and  the  intensity  R=P  -=-2.  In  any  section  of  G,  as 
XX,  let  Kf  be  the  projection  of  the  gravity  axis.  Introduce  at 
K?  two  equal  and  opposite  forces  equal  to  R  and  with  their  lines 
of  action  parallel  to  that  of  R.  Then  in  the  section  there  is  flex- 
ure stress  due  to  the  flexure  moment  Rl,  and  tensile  stress  due  to 
the  component  of  R2  perpendicular  to  the  section,  =R^.  Then 
the  maximum  stress  in  the  section  =/  +  /i. 

SIC 

T' 

A  section  may  be  assumed,  and  A,  7,  and  c  become  known; 
the  maximum  stress  also  becomes  known,  and  may  be  compared 
with  the  ultimate  strength  of  the  material  used. 

Obviously  this  resulting  maximum  stress  is  greater  when  the 
line  of  the  reaction  is  MN  than  when  it  is  KL.  Also  it  is  greater 
when  MN  is  perpendicular  to  EA  than  if  it  were  inclined  more 
toward  the  center  line  of  the  frame.  The  assumptions  therefore 
give  safety.  If  the  force  P  could  only  act  downward,  as  in  a 
steam  hammer,  KL  would  be  used  as  the  line  of  the  reaction. 

The  part  D  in  the  bolted  frame  is  not  equivalent  to  a  beam 
with  end  supports  and  a  central  load  like  C,  Fig.  263,  but  more 
nearly  a  beam  built  in  at  the  ends  with  central  load,  and  it  may 
be  so  considered,  letting  the  length  of  the  beam  equal  the  hori- 
zontal distance  from  E  to  F,  =l\.  Then  the  stress  in  the  mid- 
section  will  be  due  to  the  flexure  moment  —5-,  and  the  maximum 

o 
PI  C 

stress  =/  =  -57-.    The  values  c  and  I  may  be  found  for  an  assumed 
ol 

section,  and  /  becomes  known. 


MACHINE  FRAMES.  455 

249.  Steam-hammer  Frames. — Steam-hammers  are  made  both 
with  "open-side  "  and  "closed  "  frames.     They  may  therefore  be 
designed  by  methods  already  given,  if  the  maximum  force  applied 
is  known.     The  problem  is,  therefore,  to  find  the  value  of  this 
maximum  force. 

There  are  two  types  of  steam-hammers: 

Type  I.  Single-acting.  A  heavy  hammer-head  attached  to 
a  steam-piston  is  raised  to  a  certain  height  by  steam  admitted 
under  the  piston.  The  steam  is  then  exhausted  and  the  hammer- 
head with  attached  parts  falls  by  gravity  to  its  original  position. 
The  energy  of  the  blow  =  Wl,  where  W  is  the  falling  weight  and 
/  is  the  height  of  fall. 

Type  2.  Double-acting.  A  lighter  hammer-head  is  lifted  by 
steam  acting  under  its  attached  piston,  and  during  its  fall  steam 
is  admitted  above  the  piston  to  help  gravity  to  force  it  downward. 
The  energy  of  the  blow  =  Wl  (as  before)  plus  the  energy  received 
from  the  expansion  of  the  steam;  or,  if  the  steam  acts  throughout 
the  entire  stroke,  the  energy  of  b\ow  =  Wl  +  pAl,  where  p  is  the 
mean  pressure  per  square  inch  and  A  is  the  area  of  the  upper 
side  of  the  piston. 

250.  Stresses  in  Single-acting  Frames. — In  type  i,  when  the 
action  is  as  described,  a  force  acts  downward  upon  the  frame  during 
the  lifting  of  the  hammer.     The  intensity  of  this  force  =  pA  =the 
mean  pressure  of  steam  admitted  multiplied  by  area  of  piston, 
and  the  line  of  action  is  the  axis  of  the  piston-rod.     During  the 
fall  of  the  hammer  the  cylinder  and  frame  act  simply  as  a  guide, 
and  no  force  is  applied  to  the  frame  except  such  as  may  result 
from  frictional  resistance.     The  hammer  strikes  an  anvil  which 
is  not  attached  to  the  frame,  but  rests  upon  a  separate  foundation. 

But  a  greater  force  than  pA  may  be  applied  to  the  frame. 
In  order  that  a  cushioned  blow  may  be  struck,  the  design  is  such 
that  steam  may  be  introduced  under  the  piston  at  any  time  during 
its  downward  movement,  and  this  steam  is  compressed  by  the 


456 


MACHINE   DESIGN. 


advancing  piston.  A  part  of  the  energy  of  the  falling  hammer 
is  used  for  this  compression.  The  pressure  in  the  cylinder  result- 
ing from  this  compression  is  communicated  to  the  lower  cylinder- 
head  and  through  it  to  the  frame.  Under  certain  conditions  steam 
might  be  admitted  at  such  a  point  of  the  stroke  that  all  of  the 
energy  of  the  falling  hammer  might  be  used  in  compressing  the 
steam  to  the  end  of  the  stroke.  The  hammer  would  then  just 
reach  the  anvil,  but  would  not  strike  a  blow. 

Fig.  264,  a  shows  by  diagram  a  hammer  of  type  i.     Steam  is 
admitted,  the  piston  is  raised,  the  exhaust-valve  is  opened,  and 


(V) 


FIG.  264. 

the  piston  falls.  But  at  some  point  in  the  stroke  steam  is  again 
admitted,  filling  the  cylinder,  and  the  valve  is  closed.  Com- 
pression occurs  and  absorbs  all  or  part  of  the  energy  Wl,  In  the 
latter  case  the  hammer  will  strike  the  anvil  a  blow  whose  energy 
is  equal  to  Wl  minus  the  work  of  compressing  the  steam  in  C. 


MACHINE  FRAMES.  457 

The  compression  is  shown  upon  a  pressure-volume  diagram, 
Fig.  264,  b.  Progress  along  the  vertical  axis  from  B  toward  E 
corresponds  to  the  downward  movement  of  the  hammer.  Vertical 
ordinates  therefore  represent  space,  5,  moved  through  by  the 
hammer;  or,  since  SA  =  volume  displaced  by  the  piston,  the 
vertical  ordinates  may  also  represent  volumes.  EF  represents 

V2 
the  volume  V%  of  the  clearance  space,  or  —r,  the  piston  movement 

which  corresponds  to  the  clearance.  Horizontal  ordinates  meas- 
ured from  BF  represent  absolute  pressures  per  square  inch. 
Let  pi  represent  the  absolute  boiler  pressure  represented  by  DK. 
TN  is  the  line  of  atmospheric  pressure.  During  the  lifting  of  the 
hammer  the  upper  surface  of  the  piston  is  exposed  to  atmospheric 
pressure  and  the  lower  surface  is  exposed  to  pressure  just  suffi- 

W 
cient  to  raise  the  hammer,  =-r-.     The  work  of  lifting  is  represented 

by  the  area  NTHG.  This  work  equals  the  energy,  Wl,  which 
the  hammer  must  give  out  in  some  way  before  reaching  the  anvil 
again.  When  the  piston  has  fallen  to  some  point,  as  Z>,  steam 
may  be  let  in  below  it  at  boiler  pressure,  DK.  The  advancing 
piston  will  compress  this  steam,  and  KM  will  be  the  compression 
curve.*  The  work  of  compression  is  represented  by  the  area 
RKMN.  If  the  compression  is  to  absorb  all  the  energy  Wl,  the 
area  which  represents  the  work  of  compression  must  equal  the 
area  which  represents  Wl.  Hence  area  NTHG  must  equal 
RKMN;  or,  since  the  area  RLGN  is  common  to  both,  the  area 
RTHL  must  equal  the  area  LKMG.  The  point  at  which  com- 
pression must  begin  in  order  to  cause  this  equality  may  be  found 
by  trial.  The  greatest  unit  pressure  reached  by  compression  is 
represented  by  EM,  The  greatest  pressure,  p2,  upon  the  lower 
cylinder-head  is  represented  by  NM,  since  atmospheric  pressure 

*  Assuming  pV  =  constant. 


458  MACHINE  DESIGN. 

acts  on  the  outside.  The  corresponding  total  force  communicated 
to  the  frame  =  p2A  =P* 

If  compression  had  begun  earlier  the  energy  would  have  been 
absorbed  before  the  hammer  reached  the  anvil,  the  piston  would 
have  stopped  short  of  the  end  of  the  stroke,  the  compression  curve 
would  have  been  incomplete,  and  the  greatest  pressure  would  have 
been  less  than  EM.  Obviously  if  compression  had  begun  later 
the  greatest  pressure  would  have  been  less  than  EM.  Therefore 
the  force  P,  =  p2A,  with  the  cylinder's  axis  for  its  line  of  action, 
is  the  greatest  force  that  can  be  applied  to  the  frame  in  the  regular 
working  of  the  hammer. 

A  greater  force  might  be  accidentally  applied.  For,  suppose 
that  water  is  introduced  into  the  cylinder  in  such  quantity  that 
the  piston  reaches  it  before  the  hammer  reaches  the  anvil,  then 
all  the  energy  will  be  given  out  to  overcome  the  resistance  of  the 
water.  The  resulting  force  is  indeterminate,  because  the  space 
through  which  the  resistance  acts  is  unknown.  This  force  may 
be  very  great.  The  force  applied  to  the  frame  may  be  limited 
by  the  use  of  a  "breaking- piece."  Thus  the  studs  which  hold 
on  the  lower  cylinder-head  may  be  drilled  f  so  that  they  will 
break  under  a  force  KP,  in  which  K  is  a  factor  ot  safety  and  P 
is  the  force  found  above.  Then  the  breaking-piece  will  be  safe 
under  the  maximum  working  force,  but  will  yield  when  an  acci- 
dental force  equals  KP,  thus  limiting  its  value.  The  frame  may 
be  designed  for  a  maximum  force  KP. 

251.  Problem,  Type  i.  —  Let  W,  weight  of  hammer  and 
attached  parts,  =2000  Ibs.;  /,  maximum  length  of  stroke,  =24 
inches;  A,  effective  area  of  piston,  =50  square  inches;  clearance 
=  15  pel  cent;  boiler  pressure  =85  Ibs.  by  gauge.  Steam  is 
admitted  to  litt  the  hammer,  pressure  being  controlled  by  throttling. 

*  In  which  A  is  the  effective  area  of  the  piston,  i.e.,  area  of  the  piston  less 
area  of  the  rod. 
f  See  page  156. 


MACHINE  FRAMES.  459 

The  pressure  per  square  inch  that  will  just  lift  the  hammer  = 
2000  Ibs. -^  50  square  inches  =40  Ibs.  In  Fig.  264,  NG  repre- 
sents 40  Ibs.,  and  NT  represents  the  volume  displaced  by  the 
piston  during  a  complete  stroke.  Hence  NTHG  represents  the 
work  of  lifting  the  hammer,  or  the  energy  that  must  be  absorbed 
just  as  the  hammer  reaches  the  anvil.  Trial  shows  that  to 
accomplish  this,  compression  must  begin  at  just  about  6  inches 
from  the  end  of  the  stroke.  The  maximum  resulting  pressure, 
represented  by  NM,  equals  258  Ibs.  per  square  inch.  The  total 
pressure  acting  downward  on  the  frame  =p2^-  =  258  X  50  =  12,900 
Ibs.  =P.  If  the  factor  of  safety,  K,  is  5,  the  strength  of  the 
breaking-piece  =  KP  =  5X12, 900  =64, 500  Ibs.  This  is  the  maxi- 
mum force,  and  hence  may  be  used  as  a  basis  of  the  frame  design. 

252.  Stresses   in    Double-acting   Frames.  —  In    type    2    the 
maximum  working  force  may  be  found  by  a  similar  method.     In 
Fig.  265,  NG  represents  the  pressure  per  square  inch  of  piston 
necessary  to  raise  the  hammer.     The  area  NTHG  represents 
the  energy  stored  in  the  hammer  by  lifting.     The  area  HSJL 
represents  the  work  done  by  steam  at  boiler  pressure  acting  on 
the  upper  piston  face  while  the  piston  descends  to  D.     At  this 
point  steam  is  exhausted  above  the  piston  and  let  in  below  it,  and 
compression  takes  place  during  the  remainder  of  the  stroke.     To 
absorb  ail  the  energy  of  the  hammer  by  compression,  the  areas 
NTSJLG  and  RKMN  must  be  equal.     The  area  NRLG  is 
common  to  both;    hence  the  area  LKMG  must  equal  the  area 
RTSJ.     The  point  at  which  compression  must  begin  in  order 
to  cause  this  equality  may  be  found  by  trial. 

253.  Problem,  Type   2.  —  Let    W,    weight    of    hammer   and 
attached   parts,    =6co  Ibs.;  /,  maximum  length  of  stroke,  =24 
inches;    A,    effective    area    of   piston  (both   faces),   =50  square 
inches ;  clearance  =  15  per  cent;  boiler  pressure  =  85  Ibs.  by  gauge. 
The  construction  in  Fig.  265  shows  that  compression,  beginning 
at  gj  inches  before  the  end  of  the  piston's  stroke,  absorbs  all 


460 


MACHINE  DESIGN. 


the  energy  of  the  hammer,  and  gives  325  Ibs.  as  a  maximum 
pressure  per  square  inch.  Then  the  maximum  working  force 
=  325X50  =  l6>25°-  If  K  =  $,  the  strength  of  the  breaking-piece 
=  16,250X5  =81,250  Ibs. 


FIG.  265. 

254.  Other  Stresses  in  Hammer  Frames. — An  accidental  force 
acting  upward  may  be  applied  to  the  hammer  frame.  The  boiler 
pressure  is  necessarily  greater  than  that  which  is  necessary  to 
lift  the  hammer.*  Thus  in  §  251  a  pressure  of  40  Ibs.  per  square 
inch  is  sufficient  to  lift  the  hammer,  but  the  boiler  pressure  is 
85  Ibs.  per  square  inch.  If  the  throttle-valve  were  opened  wide 
and  held  open  during  the  movement  of  the  hammer  upward,  the 
energy  stored  in  the  hammer  when  it  reaches  its  upper  position 
would  equal  the  product  of  boiler  pressure,  piston  area,  and  length 
of  stroke,  =85X50X24  =  106,000  in.-lbs.  The  energy  necessary 

*  So  that  it  may  be  possible  to  work  the  hammer  when  the  steam-pressure  is 
lower  in  the  boiler. 


MACHINE  FRAMES.  461 

to  just  lift  the  hammer  is  40X50X24=48,000  in.-lbs.  The 
difference  between  these  two  amounts  of  energy,  =58,000  in.-lbs., 
will  exist  as  kinetic  energy  of  the  moving  hammer;  and  it  must 
be  absorbed  before  the  hammer  can  be  brought  to  rest  in  its 
upper  position.  The  force  which  would  result  from  stopping 
the  hammer  would  be  dependent  upon  the  space  through  which 
the  motion  of  the  hammer  is  resisted.  Springs  are  often  provided 
to  resist  the  motion  of  the  hammer  when  near  its  upper  position. 
These  springs  increase  the  space  factor  of  the  energy  to  be  given 
out  and  thereby  reduce  the  resulting  force.  An  automatic 
device  for  closing  the  throttle-valve  before  the  end  of  the  stroke 
and  introducing  steam  for  compression  above  the  piston  may 
be  used.  The  steam  is  then  a  fluid  spring. 

255.  Design  of  Crane  Frames. — A  crane  frame  is  to  be  de- 
signed from  the  following  specifications :  Maximum  load,  5  tons 
=  10,000  Ibs.;  radius  =  maximum  distance  from  the  line  of  lifting 
to  the  axis  of  the  mast,  =  18  feet;  height  of  mast  =  20  feet;  radial 
travel  of  hook  in  its  highest  position  =  5  feet;  axis  of  jib  to  be  15 
feet  above  floor  line.  Fig.  266  shows  the  crane  indicated  by  the 
center  lines  of  its  members. 

The  external  forces  acting  on  the  crane  may  be  considered  first. 
A  load  of  10,000  Ibs.  acts  downward  in  the  line  ab.  This  is  held 
in  equilibrium  by  three  reactions :  one  acting  horizontally  toward 
the  left  through  the  upper  support,  i.e.,  along  the  line  be;  another 
acting  horizontally  toward  the  right  through  the  lower  support, 
i.e.,  in  the  line  ad;  a  third  acting  vertically  upward  at  the  lower 
support,  i.e.,  in  the  line  cd.  The  crane  is  a  "four-force  piece." 
One  force,  AB,  is  completely  known,  the  other  three  are  known 
only  in  line  of  action.  Produce  ab  and  be  to  their  intersection  at 
M.  The  line  of  action  of  the  resultant  of  ab  and  be  must  pass 
through  M.  The  resultant  of  cd  and  da  must  be  equal  and 
opposite  to  the  resultant  of  ab  and  be,  and  must  -have  the  same 
line  of  action.  But  the  line  of  action  of  the  resultant  of  cd  and 


462 


MACHINE  DESIGN. 


da  must  pass  through  N.  Hence  M N  is  the  common  line  of 
action  of  the  resultants  of  ab  and  be  and  of  cd  and  da.  Draw 
the  vertical  line  AB  *  representing  10,000  Ibs.  upon  some  assumed 
scale;  from  B  draw  BC  parallel  to  be,  and  from  A  draw  AC  parallel 
to  MN.  The  intersection  of  these  two  lines  locates  C  and  deter- 
mines the  magnitude  of  BC.  Now  AC  is  the  resultant  of  AB 
and  BC,  and  CA,  equal  and  opposite,  is  the  resultant  of  CD 


Force  Diagram 


FIG.  266. 

and  DA.  Therefore  CA  has  but  to  be  resolved  into  vertical  and 
horizontal  components  to  determine  the  magnitudes  of  CD  and 
DA.  The  force  polygon  is  therefore  a  rectangle  and  CD=AB, 
and  BC=DA. 

From  the  forces  AD  and  CD  acting  at  N,  the  supporting 
journal  and  bearing  at  the  base  of  the  crane  may  be  designed; 
and  from  the  force  BC,  acting  at  V,  the  upper  journal  and  bearing 
may  be  designed. 

256.  Jib. — The  forces  acting  on  the  jib  are,  first,  AB  acting 
vertically  downward  at  its  end;  second,  an  upward  reaction 

*  See  Force  Diagram,  Fig.  266. 


MACHINE  FRAMES. 


463 


from  the  brace  at  H,  whose  line  of  action  coincides  with  the  axis 
of  the  brace;*  third,  a  downward  reaction  at  L  where  the  jib 
joins  the  mast,  whose  line  of  action  must  coincide  with  the  line 
of  action  of  the  resultant  of  AB  and  the  brace  reaction.  LK  is 
therefore  this  line  of  action. 

FIG.  267. 


I--"'"'               e             T 

/e            a\b 

'                            / 

/if. 


FIG.  268. 

In  the  force  diagram  draw  BE  parallel  to  the  center  line  of  the 
brace,  and  draw  EA  parallel  to  LK.  Then  BE  will  represent  the 
brace  reaction,  and  EA  will  represent  the  reaction  at  L  in  the  line 
ae.  Let  RQ,  Fig.  267,  represent  the  jib  isolated,  ae,  be,  and  ab 
are  the  lines  of  action  of  the  three  forces  acting  upon  it.  The 
vertical  components  of  these  forces  are  in  equilibrium,  and  tend 
to  produce  flexure  in  the  jib.  The  horizontal  components  are 
in  equilibrium  and  tend  to  produce  tension  in  the  jib.  The  vertical 


*  Considering  the  joint  between  the  brace  and  jib  equivalent  to  a  pin  con- 
nection. 


464  MACHINE  DESIGN. 

force  acting  at  R  is  FA*  =5000  Ibs.,  the  vertical  force  acting  at 
T  is  FB,  =15,000  Ibs.,  and  the  vertical  force  acting  at  Q  is  AB, 
=  10,000  Ibs. 

Flexure  is  also  produced  in  the  jib  by  its  own  weight  acting  as 
a  uniformly  distributed  load. 

In  order  to  design  the  jib,  standard  rolled  forms  may  be 
selected  which  will  afford  convenient  support  for  the  sheave  car- 
riage. Two  channels  located  as  shown  in  Fi^.  268  will  serve  for 
this  crane.  For  trial  12 -inch  heavy  channels  are  chosen.  From 
Carnegie's  Hand-book,  the  moment  of  inertia  for  each  channel 
about  an  axis  perpendicular  to  the  web  at  the  center =7  =  248; 
c  =  6  inches;  the  weight,  w,  of  two  channels  per  inch  of  length 
=  8Jlbs. 

The  total  weight  of  the  two  channels  =w/  =  8JXi8x  12  = 
1800  Ibs.  The  vertical  reaction,  Pl5  at  R,  Fig.  267,  due  to  this 

iwl22     wk2  \ 
weight  is  PI  =  ( 1*  12,  from  the  equation  of  moments, 

due  to  the  weight  of  jib  about  the  point  T.  Introducing  numerical 
values,  PI  =450  Ibs.  The  total  reaction  at  R  is  therefore  5000 
—  450=4550  Ibs.  A  diagram  of  moments  of  flexure  may  now  be 
drawn  under  the  jib,  Fig.  267.  Considering  the  portion  TQ,  the 

wli2 
moment    at    T  =  Pli+~  -=741,600    in. -Ibs.     Divide    TQ    into 

three  equal  parts.      At   the   division   nearest    T  the  moment  = 

U'/32 

P/3+-  —  ;  in  which /s  =  the  distance  from  Q  to  the  section  con- 
sidered. The  moment  at  the  other  two  points  may  be  found 
by  similar  method. 

The  moment  at  any  point  at  the  left  of  T  =  4550X^4  +  -  — ; 

in  which  /4  is  the  distance  from  R  to  the  section  considered.  From 
the  values  thus  found  the  diagram  of  flexure  moments  may  be 

*  See  force  diagram,  Fig.  266. 


MACHINE  FRAMES. 


drawn. 

The  resulting  fiber  stress  =  /  = 


The  maximum  value  is  at  T,  where  Mb 

^^^ 


465 

=  74I,6oo  in. -Ibs. 
=  897 1  Ibs.   The 


horizontal  component  acting  at  R  is  equal  to  FE  (see  force  dia- 
gram, Fig.  266)  =12,000  Ibs.  An  equal  and  opposite  horizontal 
force  must  act  at  T.  Between  T  and  Q  there  is  no  tensile  stress 
due  to  the  forces  AB,  BE,  and  AE. 

FIG.  269. 


FIG.  270. 

Another  force  which  modifies  this  result  needs  to  be  considered. 
Let  AB,  Fig.  269,  be  the  upper  surface  of  the  jib.  The  load  is 
supported  as  shown.  The  chain  which  is  fastened  at  B  passes 
over  the  right-hand  carriage  sheave,  down  and  under  the  hook 
sheave,  up  and  over  the  left-hand  carriage  sheave,  horizontally 
to  the  sheave  at  A,  and  thence  to  the  winding  drum.  If  a  load 
of  10,000  Ibs.  is  supported  by  the  hook  there  will  be,  neglecting 
friction,  a  tension  of  5000  Ibs.  throughout  the  entire  length  of  the 


466  MACHINE  DESIGN. 

chain  from  B  to  the  winding  drum.  There  is  therefore  a  force 
of  5000  Ibs.  tending  to  bring  A  and  B  nearer  together,  and  hence 
to  produce  compression  in  the  jib.*  The  resultant  tension  be- 
tween R  and  T  is  12,000-  5000  =  7000  Ibs.,  while  between  T  and 
Q  there  is  a  compression  of  5000  Ibs.  The  cross-sectional  area 
of  the  two  channels  selected  =  30  square  inches.  Hence  -the  unit 
tensile  stress  =  /i  =  7000-^30  =  233  Ibs,  The  maximum  unit  ten- 
sile stress  in  the  jib  =/  +  /i  =8971  +233  =9204  Ibs.  per  square  inch. 

If  the  channels  are  of  steel,  their  unit  tensile  strength  will 
probably  equal  60,000  Ibs.  per  square  inch.  The  factor  of 
safety  =  60,000^9204=6.5.  In  a  crane  a  load  may  drop  through 
a  certain  space  by  reason  of  the  slipping  of  a  link  that  has  been 
caught  up,  or  the  failure  of  the  support  under  the  load  while  the 
chain  is  slack.  When  this  occurs  a  blow  is  sustained  by  the  stress 
members  of  the  crane.  The  energy  of  this  blow  equals  the  load 
multiplied  by  the  height  of  fall.  But  the  stress  members  of  the 
crane  are  long,  and  the  yielding  is  large.  Hence  the  space  through 
which  the  blow  is  resisted  is  large  and  the  Resulting  force  is  less 
than  with  small  yielding.  In  other  words,  the  stress  members  act 
as  a  spring,  reducing  the  force  due  to  shock.  Hence,  in  a  crane 
of  this  type,  the  ductile  and  resilient  material  is  liable  to  modified 
shock,  and  a  factor  of  safety  =6. 5  is  large  enough. 

The  jib  might  also  be  checked  for  shear,  but  in  general  it  will 
be  found  to  have  large  excess  of  strength. 

257.  Mast. — Fig.  270  shows  the  mast  by  its  center  line  with 
the  lines  of  action  of  the  forces  acting  upon  it.  It  is  equivalent 
to  a  beam  supported  at  C  and  D  with  a  load  at  A.  The  moment 
of  flexure  at  A  equals  the  force  acting  in  the  line  fcf  multiplied 
by  the  distance  CA  in  inches,  =9000  Ibs.  X  60  inches  =  540,000 

*  There  is  also  flexure  due  to  this  force  multiplied  by  the  distance  from  the 
centre  line  of  the  horizontal  chain  to  the  gravity  axis  of  the  jib.  This  is  small 
and  may  be  neglected. 

f  The  force  BC  in  Fig.  266. 


MACHINE  FRAMES. 


467 


in.-lbs.  =  Mb.  The  flexure  moment  is  a  maximum  at  this  point, 
and  decreases  uniformly  toward  both  ends.  The  moment  diagram 
is  therefore  as  shown.  The  maximum  fiber  stress  due  to  this 
flexure  moment  =/  =  M ^c  -r- 1.  Selecting  two  light  1 2 -inch  channels, 

,     540000X6 

7  =  176;  c  =6  inches;  /  =  - — — — 7—  =9200  Ibs. 

2X1 7^ 

The  tension  in  the  mast  equals  the  vertical  component  of  the 
force  acting  in  the  line  ae*  =  $ooo  Ibs.  (Actually  reduced  to 
455°  by  the  effect  of  the  weight  of  the  jib.)  The  compression  in 
the  mast  due  to  the  tension  in  the  chain  =  5000  Ibs.  between  A 
and  the  point  of  support  of  the  winding  drum  B.  The  tension 
and  compression  therefore  neutralize  each  other,  except  below  B, 
where  the  flexure  moment  is  small.  Hence  the  maximum  unit 
stress  in  the  mast  is  9200  Ibs.  The  factor  of  safety  =60000  -^9200 
=  6.5,  which  is  safe  as  before.  This  also  may  be  checked  for  shear. 

258.  Brace. — The  compression  stress  in  the  brace  is  19,000 
lbs.,f  and  the  length,  19  feet,  is  such  that  is  needs  to  be  treated 
as  a  "long  column.*1  Because  of  the  yielding  of  joints  and  of 
the  other  stress  members,  the  brace  is  intermediate  between  a 
member  with  "hinged  ends"  and  "flat  ends";  therefore  for  safety 
it  should  be  considered  as  hinged.  In  the  treatment  of  long 
columns,  the  " straight-line  formula"  will  be  used.t  This 
formula  is  of  the  form 

P    *    R    rl 
j  =  p=E-  C-. 

P  is  the  total  force  that  will  cause  incipient  buckling,  and  hence 
the  force  that  will  destroy  the  column;  A  is  the  cross-sectional 
area  of  the  column;  p  is  the  unit  stress  that  will  cause  buckling; 


*  The  force  AE  in  Fig.  256. 

•\  See  force  diagram,  Fig.  266. 

J  For  discussion  of  long-column  formulae  see  "  Theory  and  Practice  of  Mod- 
ern Framed  Structures,"  by  Johnson,  Bryan,  and  Turneaure,  page  143.  Published 
by  John  Wiley  &  Sons. 


MACHINE  DESIGN. 

B  and  C  are  constants  derived  from  experiments  on  long  columns 
(the  values  of  B  and  C  vary  with  the  method  of  attachment  of  the 
ends  of  the  column,  and  with  the  material  of  the  column) ;  /  is 
the  length  of  the  column  in  inches;  and  r  is  the  radius  of  gyration 
of  the  cross-section,  =Vl+A,  I  being  the  moment  of  inertia  of 
the  cross-section  referred  to  the  axis  about  which  buckling  takes 
place. 

Values  of  B  and  C  are  as  follows: 

P  I 

For  wrought  iron,  hinged  ends,  -7=42000  — 157— . 

A  Y 

11     flat          "     --42000-128-. 

P  I 

mild  steel,       hinged     "     -r  =52500  —  220—. 

A.  Y 

"      "        "          flat          "     ^-  =  52500-179-. 

^T.  / 

• 

The  brace  will  be  of  mild  steel  channels,  and  the  encls  will  be 
considered  as  hinged.  The  formula  to  be  used  is  therefore 

P  I 

^=52500-220-, 

from  which 

/  /Nu 

P  =  { 52500  —  220—  hi. 

\  Y I 

Channel  bars  may  be  selected  and  values  of  Y  and  A  become 
known  from  tables.  For  trial  5 -inch  light  channels  are  chosen. 
Carnegie's  tables  give  for  2  channels,  ^=1.95  inches,  and  A  = 
3.9  square  inches.  Introducing  these  values  in  the  above  equation, 
with  /=i9'Xi2  =  228",  gives  P=  106,970  Ibs.  Since  the  maxi- 
mum compression  force  sustained  by  the  brace  =19,000  Ibs., 


MACHINE  FRAMES. 


469 


the  factor  of  safety  =  106,970^19,000  =  5.6  +  .  This  is  a  smaller 
value  than  those  for  the  jib  and  mast,  but  it  is  probably  inadvisable 
to  use  larger  channels  because  of  convenience  in  making  the 
connection  with  the  jib  and  mast. 

But  the  brace  must  be  made  safe  against  side  buckling.  The 
two  channels  may  be  considered  as  acting  as  a  single  member  if 
they  are  braced  laterally.  The  lateral  bracing  will  be  determined 
later.  In  Fig.  271  the  moment  of  inertia  about  the  axis  X  for 


FIG.  271. 


FIG.  272 


each  channel  =  7  (from  table.)  If  the  moment  of  inertia  of  each 
about  the  axis  Y  be  made  =  7,  the  radius  of  gyration  will  be  the 
same  about  both  axes,  the  values  in  the  above  equation  will  be 
the  same,  and  there  will  be  the  same  safety  against  side  buckling 
as  against  buckling  in  the  plane  through  the  axis  of  the  mast. 
Therefore  it  is  only  necessary  to  make  the  distance,  0,  of  each 
channel  from  the  axis  of  Y  such  that  Iy  =  7-  The  moment  of 
inertia  of  one  channel  about  its  own  gravity  axis  GG  =0.466.  Its 
moment  of  inertia  about  Y  =  7  =Ic+Ax2.  Solving,  x2  =  (I 
+A,  whence 

7-0.466 


X2= 


1.828. 


Hence  the  distance  apart  of  the  gravity  axes  =  1.828X2  =3.65. 
But  the  gravity  axis  is  0.44  inch  from  the  face  of  the  web,  i.e., 
x—  a  =0.44.  Therefore  the  distance,  b,  between  the  channels 
=  3-65  —  0.88  =  2.77  inches.  Convenience  in  construction  would 


470  MACHINE  DESIGN. 

undoubtedly  dictate  a  greater  distance,  and  hence  greater  safety 
against  side  buckling. 

The  position  of  these  two  channels  relative  to  each  other  must 
be  insured  by  some  such  means  as  diagonal  bracing.  See  Fig.  272. 
The  distance,  /,  must  be  such  that  the  channels  shall  not  buckle 
separately  under  half  the  total  load.  Solving  the  long-column 
formula  gives 


in  which  P  is  the  load  sustained  by  each  channel  (  =  19000-4-2  = 
9500  Ibs.)  multiplied  by  the  factor  of  safety,  say  6.  The  radius  of 
gyration,  r,  is  about  a  gravity  axis  parallel  to  the  web,  =  Vl  +A  = 
V  0.466  -4--  1  .95  =  0.488. 


/ 
= 
\ 


9500X6^0.488 

=    52500-^—       -    --  =  51.8. 
\  1-95     /  22° 


The  value  of  /,  therefore,  must  not  be  greater  than  51.8  inches 
but  it  may  be  less  if  convenience,  or  the  use  of  standard  braces 
requires. 

259.  Crane  Frame  with  Tension  Rods.  —  The  brace  in  the 
crane  just  considered  may  be  replaced  by  tension  rods,  as  shown 
in  Fig.  273.  This  allows  the  load  to  be  moved  radially  through- 
out the  entire  length  of  the  jib.  The  force  polygon,  Fig.  274, 
shows  tension  equal  to  37,000  Ibs.  in  the  tension  rods,  and  com- 
pression in  the  jib  =  35,  800  Ibs.  If  the  tension  rods  are  made  of 
mild  steel  with  an  ultimate  tensile  strength  of  60,000  Ibs.,  and 
a  factor  of  safety  =6,  the  cross  -sectional  area  must  equal 
37,000X6-^-60,000=3.7  square  inches.  If  two  rods  are  used  the 
minimum  diameter  of  each  =  1.535  »  sav>  x&  inches. 

The  mast  is  a  flexure  member  20  feet  long  supported  at  the 
ends,  and  sustaining  a  transverse  force  of  35,800  Ibs.  at  a  distance 
of  5  feet  from  the  upper  end.  The  upper  end  reaction  is  there- 


MACHINE  FRAMES. 


471 


fore   35, 8ooXH  =  26,850   Ibs.,  and    the   maximum   moment   of 
flexure    at    ^  =  26,850X60  =  1,611,000    in. -Ibs.     Selecting    two 

1 5- inch  heavy  eye-beams,  7  =  750X2;  c  =  7i;   •'•/  =  —        — — - 

750x2 

=  8055  Ibs.  =  maximum  unit  stress  in  the  mast.     The  factor  of 

60000 
safety  =8        =7.4,  safe. 


a\b 

I 10000 J 


*"*• 


d 


9000*   A 


35800' 


35800 i 

FIG.  274. 

The  moment  of  flexure  in  the  jib  is  a  maximum  when  the 
maximum  load  is  suspended  at  its  center.     The  maximum  flexure 

PI    wl2 

moment  due  to  the  load  and  the  weight  of  the  jib  —Mb  =  —  +  -5-  , 

4        o 

in  which  P  =  load  =  10,000  Ibs. ;  w  =  weight  of  two  channels  per 
foot  of  length,  =100  Ibs.  if  1 2-inch  heavy  channels  are  chosen. 
Substituting  numerical  values,.  Mb  =  49,050  ft. -Ibs.  =588,600  in.- 

Ibs.     The  resulting  maximum  fiber  stress  due  to  flexure,  /  =  — ~ 


588600X6 
2X248 


7120  Ibs. 


472  MACHINE  DESIGN' 

Compression  in  jib  due  to  chain  tension  =  10,000  Ibs.; 
Compression  in  jib  due  to  load  =35,800  Ibs.; 
Combined  compression  due  to  both  =  10,000X35, 800  =45, 800  Ibs. 
Unit  compress  ion  =  combined  compression  -f-  area  of  the  two 

channels  = =  1526  Ibs. 

30 

Maximum  fiber  stress  due  to  combined  flexure  and  com- 
pression =71 20 +1526  =8647  Ibs.  The  factor  of  safety  =  — 

0040 

=  6.96.     If  a  smaller  factor  of  safety  were  desired,  smaller  channels 
could  be  used.     The  jib  may  be  checked  for  shear. 

The  load  might  be  moved  nearly  up  to  the  mast,  hence  the 
joint  at  F  must  be  designed  for  a  total  shear  of  10,000  Ibs.  The 
pin  and  bearing  at  G,  as  well  as  the  supporting  framework  for 
the  bearing  must  be  capable  of  sustaining  a  lateral  force  =  BC 
=8950  Ibs.  in  any  direction.  The  pivot  and  step  at  H  must  be 
capable  of  sustaining  the  lateral  force  AD  =  8950  Ibs.,  as  well  as 
a  vertical  downward  thrust  of  10,000  Ibs.  +the  weight  of  the  crane. 

260.  Pillar-crane  Frame. — Fig.  275  shows  an  outline  of  the 
frame  of  a  pillar  crane.  HN  represents  the  floor  level;  HK 
represents  the  pillar,  which  is  extended  for  support  to  L',  KM 
represents  one  or  more  tension  rods;  MH  represents  the  brace. 
The  load  hangs  from  M  in  the  line  ab.  The  pillar  is  supported 
horizontally  at  H,  and  vertically  and  horizontally  at  L.  The 
force  polygon  shows  the  horizontal  forces  at  H  and  L  =  30,000  Ibs., 
and  the  vertical  force  at  L=  10,000  Ibs.  From  these  the  sup- 
ports may  be  designed.  These  supports  should  provide  for 
rotary  motion  of  the  crane  about  KL,  the  axis  of  the  pillar. 
The  brace  may  be  treated  as  in  the  jib  crane,  the  compressive 
force  being  21,400  Ibs.  The  tension  rods  may  be  designed  for 
the  force  15,800  Ibs. 

The  forces  sustained  by  the  pillar  are  as  follows  (see  Fig. 
276):  FE,  the  horizontal  component  of  ^£,  =  15,000  Ibs.,  acts 


MACHINE  FRAMES. 


473 


horizontally  toward  the  right  at  K,  and  EF,  the  horizontal 
component  of  EB,  =  15,000  Ibs.  acts  toward  the  left  at  H.  BC  = 
30,000  Ibs.  acts  toward  the  left  at  H,  and  DA  =30,000  Ibs.  acts 
toward  the  right  at  L.  CD  acts  upward  at  L,  producing  a  total 
compression  in  the  portion  LH  of  10,000  Ibs.  The  force  AF  = 
5000  Ibs.  acts  to  produce  tension  between  H  and  K.  From 
these  data  the  pillar  may  be  designed  by  methods  already  given. 

FIG.  275. 


FIG.  276. 

261.  Frame  of  a  Steam  Riveter. — Let  Fig.  277  represent  a 
steam  riveter.  Both  the  frame  and  the  stake  are  acted  upon  by 
three  parallel  forces  when  a  rivet  is  being  driven.  The  lines  of 
action  of  these  forces  AB,  BC,  and  CA,  are  ab,  be,  ca.  The 
force  AB  required  to  drive  the  rivet  =  35,000  Ibs.  BC  and  CA 
may  be  found,  the  distances  EH  and  HG  being  known.  The 
moment  of  flexure  on  the  line  ca  =  35,000  Ibs.  X  74"  =  2, 590, ooo 
inch-pounds.  Let  the  line  HF  represent  this  moment.  The  moment 


474 


MACHINE  DESIGN. 


in  any  horizontal  cross-section  may  be  found  from  the  diagram 
EFG.  Any  section  of  the  frame  or  stake  may  therefore  be  checked. 
The  stake  needs  to  be  small  as  possible  in  order  that  small  boiler 
shells  and  large  flues  may  be  riveted.  In  order  that  it  may  be 
of  equal  strength  with  the  cast  iron  frame,  it  is  made  of  material 
of  greater  unit  strength,  as  cast  steel. 


72000^ 


FIG.  277. 


The  two  bolts  which  hold  the  frame  and  stake  together  sus- 
tain a  force  of  107,000  Ibs.  The  force  upon  each  therefore  is 
53,500  Ibs.  If  the  unit  strength  of  the  material  is  50,000  Ibs., 
and  the  factor  of  safety  is  6,  the  area  of  cross  -section  of  each  bolt 


would  be  = 


6X53500 
50000 


6.42  square  inches.      The  diameter  corre- 


sponding =2. 86  inches.      A  3j-inch  bolt  has  a  diameter  at  the 


MACHINE  FRAMES.  475 

bottom  of  the  thread  =  2.  88  inches,  and  and  hence  3J-inch  bolts 
will  serve  as  far  as  strength  is  concerned.  But  the  body  of  the 
bolt  is  60  inches  long,  and  each  inch  of  this  length  will  yield 
a  certain  amount,  and  the  total  yielding  might  exceed  an 
allowable  value,  even  if  a  safe  stress  were  not  exceeded.  The 
yielding  per  inch  of  length,  or  the  unit  strain  =  unit  stress  -r- 
coefficient  of  elasticity,  or 

l=-L,  but/—  =^  =  644°  Ibs. 

and  £  =  28,000,000 
644° 


28000000 


.     u 
=  .00002  3  inch. 


Total  yielding  =  ^X6o  =  .ooi38  inch. 

This  amount  of  yielding  is  allowable  and  therefore  two  3j-inch 

bolts  will  serve. 


APPENDIX. 


THE  following  method  of  determining  the  position  of  the 
slider-crank  chain  corresponding  to  the  maximum  velocity  of  the 
slider  is  largely  due  to  Professor  L.  M.  Hoskins. 

Refer  to  Fig.  278. 


FIG.  278. 

=  5  =  connecting-rod  length,  to  scale. 
=  a  =  crank  length,  to  scale. 
OA=y. 


OB  =  d. 

Angle  AOB  =90°. 


£  =  20  =  length  of  stroke  of  slider. 

From  our  study  of  the  velocity  diagram  of  the  slider-crank 
chain  (see  §  22)  we  know  that  the  length  y  will  represent  the 


477 


478  APPENDIX. 

velocity  of  the  slider  on  the  same  scale  as  the  length  a  repre- 
sents the  velocity  of  the  center  of  the  crank-pin.  The  length 
y  is  determined  by  erecting  at  O  a  perpendicular  to  the  line  of 
action  of  the  slider  and  cutting  this  perpendicular  by  the  con- 
necting-rod b,  extended  if  necessary. 

Our  problem,  then,  is  to  find  the  position  of  the  mechanism 
corresponding  to  the  maximum  value  of  y. 

Consider  the  triangle  whose  sides  are  y,  x,  and  a.  Calling 
the  angle  included  between  x  and  y,  a, 

a2=y2+x2  —  2xy  cos  a; (i) 

but  cos  or  =  -^-7 (2) 


(3) 

Clearing  (3)  of  fractions  and  transposing, 


2xy2=x3+x2b+y2x+y2b-a2x-a2b.     ...     (4) 
Differentiating, 
dy 


Transposing, 

/?Ai 

-a2-2y2.  .    .     (5) 


APPENDIX.  479 

dy 

For  maximum  value  of  y,  -r  =o;  hence  we  may  write  o  for 

the  left-hand  term  of  (5). 

— a2  —  2y2', 
-a2 (6) 

Adding  x2  and  subtracting  a2  from  both  sides  of  (6), 

x2+y2  —  a2  =  4X2 -{- 2bx  —  20? (7) 

From  (3), 

2Xy2 


Substituting  this  value  in  (7), 


(8) 

Substituting  in  (8)  the  value  of  y2  given  in  (6), 

2x($x2  +  2bx  —  a2)=  (^x2  +  2bx  —  2  a2)  (x  +  b)  ; 
.*.  6JC3  +  4&,v2  —  2  a2x  =  4.x3  +  qbx2  +  2  bx2  +  2  b2x  —  2  a2x  —  2  ba?  ; 


=o  .......     (9) 

Dividing  (9)  by  a3  and  transposing, 
a2    x     x2    x3 


Equation   (10)   gives  us  the  relation  existing  between  a,  b, 
and  x  for  the  maximum  velocity  of  the  slider. 


480  APPENDIX. 

By  taking  a  series  of  values  of  7-  and  solving  (10)  for  the 
corresponding  values  of  T-,  Curve  A  has  been  constructed.     Ordi- 


x 


nates  are  i~,  abscissae  are  T-. 
0  0 

For  any  given  problem  the   values  of  a  and  b  are  known. 

a 

Solve  for  7-. 
0 

From  Curve  A  find  the  value  of  r-  corresponding  to  this  value 


/*/• 

From  the  determined  value  of  7-  and  the  known  value  of  b 

b 

the  numerical  value  of  x  is  found. 

But  equation  (6)  gives  for  the  maximum  value  of  y  the  rela- 
tion 

—  a2, 


which  we  can  readily  solve  for  y  since  all  of  the  right-hand  terms 
are  now  known. 

AOB  being  a  right-angled  triangle, 


The  values  of  the  right-hand  member  being  known  we  can 
readily  solve  this  for  d. 

Let  m  represent  the  distance  moved  through  by  the  slider 
from  the  beginning  of  the  stroke,  then 

m  =  0  +  a— d. 


APPENDIX.  481 

The  portion  of  the  stroke  accomplished  by  the  slider  at  the 
time  of  its  maximum  velocity  expressed  as  a  fraction  of  the 
whole  stroke,  2  a, 

m 


Curve  B  shows  the  relation  between  r-  and  —  .     From  this 

0  20 

curve  we  can  see  at  a  glance  for  any  given  value  of  7-  what  per 
cent  of  the  slider's  stroke  is  accomplished  when  its  position  of 
maximum  velocity  is  reached.  Abscissae  are  values  of  r-;  ordi- 

m 
nates   —  . 


-  =  ratio  of  the  maximum  velocity  of  the  slider  to  the  velocity 
of  the  center  of  the  crank-pin.  Curve  C  shows  the  relation 

between  the  values  of  7-  and  —  .    Abscissae  are  values  of  7-  ;  ordi- 
b  a  b  ' 

y  y 

nates  —  —  i.      Add  unity  to  the  ordinates  for  actual  values  of  —  . 
a  a 

To  find  the  values  of  /?  corresponding  to  the  maximum  velocity 
of  the  slider  we  have  the  three  sides  of  the  triangle  OMB,  namely, 
by  a,  and  d.  Let 

b  +  a+d 

5=  -  . 


Then  cos  J/?  =  — j— — ,  from  which  we  can  readily  get  the 

value  of  /?. 

Curve  D  is  plotted  with  values  of  /?,  in  degrees,  as  ordinates 

d 
and  values  of  7-  as  abscissae. 


482 


APPENDIX. 


TABLE  XL. — ASSUMED  VALUES  OF  — ,  AND  CORRESPONDING  COMPUTED  VALUES 

0 

OF  -r,  T-,  — ,  — ,  AND  8,  TO  PLOT  CURVES  A,  B,  C,  AND  D 
o    b     2a    a 


X 

~b 

O.OIO 

0.015 

0.025 

0.035 

0.050 

0.075 

0.  100 

0.2OO 

a 
~b 

o  .  1005 

0.1234 

o.  1600 

o.  1902 

0.2289 

0.2832 

0.3302 

0.4817 

y 
b 

O.IOI 

0.1243 

o.  1621 

o.  1936 

0.2348 

o  .  2944 

0-3479 

0.5367 

m 

2d 

Q.4751 

o  .  4704 

0.4622 

0.4561 

o  .  4484 

o  .  4402 

o  .  4320 

0.4239 

y_ 

a 

1.005 

1.0073 

1.0131 

I.OI79 

1.0258 

1-0395 

1-0533 

I.II39 

p_ 

89°  55' 

89°  49' 

89°  44' 

89°  35' 

89°  26' 

88°  50' 

88°  1  6' 

85°   17' 

X 

o   300 

o  400 

o  .  500 

o  .  600 

O    7OO 

0.800 

O  .  OOO 

1~ 

w  .  jww 

W  .  £J-WW 

*J  .  /ww 

d 

o.  6025 

o.  7043 

o  .  7006 

0.8626 

O    O2O3 

o  .  963^ 

o   ono  c 

b 

v^  .   j\stu 

/  y 

w  .  y^w,) 

w  •  yyj 

y 

o.  7121 

0.8854 

i  .0607 

I  .  2T.QT. 

I  .422^ 

i  .  6100 

i  .  802  ; 

b 

oyo 

i.  .  ^.—  A^ 

m 

o  4273 

o  44.00 

O    4.^07 

O    4OOI 

o  .  6044 

o  .  7221 

2d 

W  •  T"*  /  O 

w  .  ^.^wvy 

•*  •  tjy  1 

w  .  ^f  vyw  ± 

' 

y. 

I  .  1819 

1  .  2tJ7l 

I     34l6 

I     4367 

I    671  3 

i  .8198 

d 

B 

D  / 

76°  41' 

J.    .  ^L^  A  W 
71°     W' 

»  •  tow/ 

65°   19' 

57°  co' 

-1  •  W  *O 

48°  2l' 

35°  8' 

r 

f  \J      «l-  J. 

1           OO 

j  /    jv 

^**          —  * 

O  J 

APPENDIX. 


483 


INDEX. 


PAGE 

Accelerated  and  retarded  motion,  cams  for 54 

Acceleration,  diagrams  of 68,  75,  79 

General  method  diagram 79 

Adaptation  in  design. ix 

Addendum 349,  359 

AHARA,  E.  H 316 

ALFORD'S  Bearings 244,  247 

American  Journal  of  Science 213 

American  Machinist 122,  125,  173,  179,  217,  229,  266,  269,  280,  321, 

344,  372,  412,  432,  442 
American  Society  of  Mechanical  Engineers. 

See  Journal;  Transactions;  also  Boiler  Code. 

ANDREWS,  E.  S 442 

Angle  of  action 356 

Annular  gears 362 

Appearance  in  design xii 

Application,  machinery  of 4 

Arc  of  action 356,  374,  375 

ARCHBUTT  and  DEELEY'S  Lubrication  and  Lubricants 252 

Axle  design 194 

Axles,  shafts  and  spindles 194-209 

BACH,  PROF.  C 104,  124,  198,  284,  359,  378,  380 

BACH,  PROF.  C.  and  E.  ROSER 416 

Ball  Bearings.     See  Roller-  and  Ball-bearings. 

BARLOW'S  cylinder  formula 103 

BARNARD,  W.  N 1 20 

BARR,  JOHN  H 120, 191 

EARTH,  CARL  G 301,  344 

BAUSCHINGER,  J 83 

Beams,  Tables  of 102 

Bearing  Pressure,  allowable  for  journals 230,  231,  242,  244,  247 

Allowable  for  roller-  and  ball-bearings 269 

Allowable  for  sliding  surfaces 191 

Allowable  for  thrust-bearings 242,  244,  247 

485 


486  INDEX. 

PAGE 

Bearings 247-260 

See  also  Journals;  Roller-  and  Ball-bearings. 

Belts 286-310,  321 

Coefficient  of  friction  of 301 

Cone  or  stepped  pulleys  for 292 

Creep  and  slip  of 301 

Crowning  pulleys  for 292 

Design,  theory  of 295 

Distances  between  shafts  for 310 

Driving  capacity,  variation  of 307 

Dynamo-belt  design 305 

Efficiency  of 302 

Intersecting  axes 290 

Pump-belt  design 304 

Shifting,  principle  of 289 

Size  of  pulleys  for 310 

Steel 321 

Transmission  of  motion  by 286 

Twist 289 

Weight  of  leather 300 

See  also  Chains;  Rope  transmission. 

Bending  moments  of  beams 102,  103 

BENJAMIN,  PROF.  C.  H 343 

Bevel-gears 382 

Skew 389 

BIRD,  PROF.  W.  W 301 

BIRNIE'S  cylinder  formula 103 

Boiler  code 112.  129,  138 

Shell,  design  of 131 

Bolts  and  screws 139-167 

Analysis  of  screw  action 142-145 

Calculation  of  bolts  subject  to  elongation . 152 

Calculation  of  screws  for  transmission  of  power , 160 

Calculation  of  screws  not  stressed  in  screwing  up 145 

Calculation  of  screws  stressed  in  screwing  up 146 

Cap  screws 140 

Classifications  and  definitions 139-150 

Coefficient  of  friction  of 162 

Design  of  bolts  for  shock 156 

Efficiency  of 162 

Lock-nuts 159 

Set-screws 140,  17? 

U.  S.  standard  threads 140 

Wrench  pull 151 

Box  pillar.     See  Supports. 


INDEX.  487 

PAGE 

Boxes 247-260 

See  also  Journals. 

Brackets 95 

See  abo  Supports. 

Brakes,  band  and  block 323 

Brass 88,  99 

See  also  Boxes;   Bronze. 

Bronze,  manganese,  naval  phosphor,  and  Tt  bin 88,  99 

See  also  Gun-metal. 

BROWN  &  SHARPE  MANUFACTURING  Co 364,  366,  369,  374,  398,  413 

BROWN  HOISTING  AND  CONVEYING  MACHINE  Co 322 

BROWNE,  D.  H 200 

BRUCE,  R.  A 417 

BUCHNER,  K 378 

Cams , 51-61 

CARPENTER,  R.  C 158 

Cassier's  Magazine 342 

Cast-iron 88,  91,  98,  99 

Parts 96 

Centre 11-13 

Location  of I6-i7,  18,  22-24 

Centrode 13-14 

Centres  of  three  links 17-18 

Chains,  Block 326 

Efficiency  of 326 

Flat  link 326 

MORSE 327 

Motion ; 15 

Open  link 325 

RENOLD 327 

Roller 326 

Stud  link 325 

CHURCH,  PROF.  I.  P 239 

Circular  plates,  stresses  in 104 

CLAUZEL,  M.  LE  BARON 121 

CLAVERINO'S  cylinder  formula 103 

Clearance,  Ball-bearir.g 272 

Gear  tooth 349 

Clutches 279-285 

See  also  Couplings. 

Columns,  long,  formulae 103,  200 

CONANT,  D.  J iii 

Connecting-rod,  angularity  of 30-32 

Constrained  motion 5-8 


488  INDEX. 

» 

PAGE 

Copper 88,  98,  99 

Cotters 176 

Couplings  and  clutches 274-285 

Band 284 

Claw,  jaw,  or  toothed 279 

Combination,  friction  and  claw 284 

Compression 275 

Denned 274 

Disengaging 279-285 

Flange 275 

Flexible 278 

Friction 280 

HALL'S 278 

HOOKE'S 277 

Hydraulic 285 

Magnetic 285 

OLDHAM'S 277 

Permanent 274-279 

Pneumatic 285 

Self-sustaining 282 

SELLER'S 276 

Sleeve,  muff,  or  quill 274 

WESTON  friction 283 

Cranes,  problems  in  design  of 461-473 

Crank-pin,  design  of 232,  236 

CRESSON,  THE  GEO.  V.  Co 313 

Critical  speed  of  shafts 204 

Cross-head  pin,  design  of 237 

Cross-head,  slotted 20-21 

CURTIS  turbine  step-bearing 244 

CUTTER,  L.  E iii,  374 

Cutting  Speeds 32 

Cycloidal  gears 353~358,  360,  373,  383 

Cylinder  formulae 103 

Heads,  stresses  in 104 

Deflections,  beams 102 

Line-shafts ... 203 

DEWRANCE,  J 224,  254 

Dynamo,  belt-design  for 288,  305 

Economy  in  design x-xii 

Efficiency  of  machines 2 

Elastic  limit,  table  of  values 99 

Elasticity,  moduli  of 99 


INDEX.  489 


Electric  conductivity 98 

Elements,  pairs  of  motion ,. 14 

Size  of 19 

Elliptical  plates,  stresses  in 104 

Energy,  definition  of i 

In  machines 2,  62-80 

Law  of  conservation  of i 

Sources  of 3 

Engineering  (London) 180 

Engineering  News 285 

Engineering  Record 247 

EWING,  PROF.  J.  A 338 

Factor  of  safety,  discussion  of 82 

For  gear  teeth 374 

Fastenings.     See  Riveted  Joints;  Bolts  and  Screws;  Keys;  Cotters;  Fits 

and  Fitting. 
Feathers.     See  Keys. 

FELLOW'S  stub-tooth  gears 375 

Fits  and  Fitting 169,  178-183 

Flanged  plates,  stresses  in 104 

Flat  plates,  stresses  in 104 

FLATHER,  J.  J 311,315,316 

Fly-wheels 328~345 

Construction  of 341 

Design,  general  method 329 

HAIGHT'S  joint  for 345 

Pump 333 

Punching  machine 329 

•Steam-engine 336 

Stresses  in  arms 339 

Stresses  in  rims 336 

Theory  of 328 

Followers 51-61 

Force,  definition  of i 

Force-fits 178 

Form,  dictated  by  stress 81,  89 

Frames 92~9S,  44i~47S 

Closed 452 

Cranes 461 

Open-side 441 

Punching-machine 442 

Riveting-machine 473 

Slotting-machine > 447 

Steam-engine,  center-crank -. 452 


490  INDEX. 


Frames  (continued). 

Steam-engine,  side-crank 449 

Steam  hammer 455 

See  also  Supports;   Machine  parts. 

Friction  brakes 323 

Clutches 280 

Coefficient   for    belts 302 

brakes  and  clutches 282 

cotters 177 

dry  surfaces 212 

force  fits 183 

lubricated  surfaces 213,  216 

roller-  and  ball-bearings 273 

ropes 311 

screws 162 

wire  ropes 3  20 

Energy  loss  by 2 

Gear  tooth 37§ 

Heat  generated  by,  in  journals 220 

Laws  of,  dry  surfaces 212 

,  imperfectly  lubricated  surfaces 213 

,  perfectly  lubricated  surfaces 214 

See  also  Journals;  Lubrication;   Sliding  Surfaces. 

FRITZ,  JOHN 343 

Gearing.     See  Toothed  Wheels. 

GENERAL  ELECTRIC  Co 230 

GLEASON,  ANDREW 376 

Works 386 

GRANT'S  Teeth  of  Gears 369 

Graphite,  as  lubricant 259 

GRASHOF'S  cylinder  formula 103 

Grasshopper  motion 47 

GREEN,  B.  M iii 

GREENHILL,  A.  G 201 

GUEST'S  maximum  shear  theory 198 

Guides,  see  sliding  surfaces 185-193 

Gun-metal 88,  99 

HALSEY,  F.  W 179,  39°,  398>  432,  435 

Harmonic  motion,  cams  for 54 

HESS,  HENRY 270 

Higher  pairs 14 

HILL,  T 266 

HIRN,  G 219 


INDEX. 


491 


,  PAGE 

HOSKINS,  PROF.  L.  M 445,  452,  477 

Hubs,  size  of 344,  370 

Stresses  in 1 79,  183 

HUNT,  THE  C.  W.  Co 3iS 


Illinois,  University  of,  Bulletin 127,  173,  325 

Indicator  pencil  mechanisms 42-46 

CROSBY 46 

TABOR 45 

THOMPSON 44 

Instantaneous  center 10-12 

Instantaneous  motion 10 

General  solution  for 30 

Involute  gears 358,  361,  373,  383 

Iron.     See  Cast-iron;  Wrought-iron. 


JANNEY,  R 269 

Jib-crane 461 

JOHNSON,  BRYAN,  and  TURNEAURE'S  Theory  and  Practice  of  Modern  Framed 

Structures 467 

JONES'  Machine  Design 344 

Journal,  Amer.  Soc.  of  Mech.  Engs 205 

Amer.  Soc.  of  Naval  Engs 1 26 

Franklin  Institute 81 

Journals 210-260 

Allowable  bearing  pressure 230,  231,  242,  244,  247 

Calculation  of,  for  strength 232 

Crank-pin  of  engine 236 

Cross-head  pin  of  engine 237 

Design  of,  by  heat  balance 224 

Friction  of 211 

General  discussion  of 210 

Heating  of 220 

LASCHE'S  experiments  on 219 

Lubrication  of 211,  252 

Main,  of  engine 233 

Materials  for,  and  bearings 231 

MOORE'S  experiments  on 217 

Proportions  of 231 

STRIBECK'S  experiments  on 219 

Thrust 239 

TOWER'S  experiments  on 217 

See  also  Roller-  and  Ball-bearings. 


492  INDEX. 

PAGE 

KENERSON,  W.  H 417 

KENNEDY,  SIR  A.  B.  W 13,  19,  50,  107,  427 

Keys 168-177 

Classification  of 168 

Cotters.. 176 

Feathers 174 

KERNOUL  and  BARBOUR 172 

Parallel 168 

Roller  ratchet 1 73 

Round  taper 175 

Saddle,  flat  and  angle 172 

Splines 1 74 

Strength  of 173 

Taper 169 

WOODRUFF 170 

Keyways,  effect  of,  on  strength 173 

KlMBALL,  A.  S 213 

KINGSBURY,  ALBERT 161,  193,  246 

LANZA,  PROF.  C 177,213,  301 

LASCHE,  W. 219,  220,  223,  378,  379 

Lathe,  bed 95,  188 

Supports 437 

LE  CONTE,  J.  N 72,  398 

Legs.     See  Supports. 

Lever-crank  chain,  location  of  centres 19 

Velocity  diagram  of 28 

LEWIS,  WILFRED 213,  296,  301,  372,  373 

Linear  velocity 24 

Points  in  different  links 28 

Line-shafts •. 202 

Linkage,  definition  of 15 

Compound 16 

Simple 16 

Lock-nuts , 159 

LOF,  E.  A 209 

Long  columns 90,  103,  200 

Lower  pairs 14 

Lubrication  of  helical  surfaces 162 

of  roller-  and  ball-bearings 272 

of  rotating  surfaces 211,  252 

of  sliding  surfaces 192 

McCoRD's  Kinematics 369 

Machine,  cycle 2 


INDEX.  493 

PAGE 

Machine,  definition  of i 

Efficiency  of 2 

Frames,  dictated  by  stress 92-95 

See  also  Frames. 

Function  of 3 

Parts,  forms  of  cast  members 96-97 

Parts,  proportions  dictated  by  stress 81-104 

Machinery 209,  376 

Main  journal,  design  of 233 

Master  Steam  Boiler-makers  Assn in 

Materials,  Tables  of  properties  of '.  88,  98-104 

MAW'S  Modern  Practice  in  Marine  Engineering 252 

Mechanism,  definition  of 16 

Location  of  centres  in  compound 22 

Mechanisms,  quick-return 32-40 

Melting  points  of  metals 98 

MERRIMAN,  PROF.  M 199,  200 

Metals,  tables  of  physical  properties  of 98 

MILLER,  SPENCER 313 

MITCHEL,  A.  G.  M 246 

Modulus  of  elasticity 99 

of  rigidity 99 

of  rupture 99 

of  section,  plane  and  polar 100-101 

Moments  of  inertia 100-101 

MOORE,  PROF.  H.  F 127,  173,  217 

MORLEY,  PROF.  A 180 

MORSE  CHAIN  Co 327 

Moss,  SANFORD  A 103,  183,  338 

Motion,  chains 15 

Constrained 5-8 

Definition  of i 

Elements 14,  19-20 

Free 4 

Helical 9 

Instantaneous n,  30 

Kinds  of,  in  machines 8-9 

Plane 8 

Relative 9 

Spheric 9 

Pairs  of  elements 14 

Parallel  motions.     See  Straight-line  motions 4I~S° 

Parallelogram 41 

Passive  resistance 5-8 


494  INDEX. 

PAGE 

PEAUCELLIER  link 4g 

PEDDLE,  PROF.  J.  B 432 

PEDERSON,  AXEL 229 

PETROFF'S  Neue  Theorie  der  Reibung 216 

Philosophical  Transactions 216,  252 

Pillar  Crane 472 

Pitch  arc 356,  374,  375 

Pitch,  axial 417 

Circular  or  circumferential 349 

Diametral 349 

Normal 391 

Pitch  circle 348 

Planing-machine,  bed 94 

Lubrication  of 193 

Table 93 

Plates,  stresses  in  flat,  flanged  and  stayed 104 

PLYMOUTH  CORDAGE  Co 315 

POIFSON'S  ratio,  table  of  values  of 99 

PORTER,  C.  T 232 

Power 278,  344 

PRATT,  C.  R 266,  271 

Pressure-vessel  walls,  formulae  for 103-104 

Prime  mover 3 

Proceedings  Amer.  Soc.  Civil  Engs 313 

Inst.  Civil  Engs 224,  254 

Inst.  Meek.  Engs 107,  in,  117,  121,  217,  240,  242,  252,  255,  417 

Master  Car  Builder's  Assn 196 

Phila.  Engs.  Club 372 

Pulleys,  cone 292 

Crowning 292 

Idler  or  guide 291 

Proper  size  of,  for  belts 310 

Proportions  of 344 

Stresses  in  arms 339 

Stresses  in  rims 336 

See  also  Fly-wheels. 

Pump,  belt  design  for 304 

Fly-wheel  design  for 333 

Punching  machine,  fly-wheel  design  for 329 

Frame  design  for 442 

Quick-return  mechanisms 32~4o 

Lever-crank  quick-return 34~37 

Slider-crank  quick-return 32-34 

WHITWORTH  quick-return 37~4o 


INDEX.  s    495 

PAGE 

Radii  of  gyration,  table  of 100 

Railway  Machinery 196 

RANKINE'S  Machinery  and  Millwork 50 

Rectangular  plates,  stresses  in 104 

Repetitive  stresses.     See  Stresses,  variable. 

REULEAUX'S  Constructor 196,  241,  315,  318,  390,  408,  435 

REYNOLDS,  PROF.  OSBORNE 216,  252 

RICHARDS,  JOHN 81,  92,  168,  175 

Rigid  body 10 

Rigidity,  modulus  of 99 

RISDON  IRON  WORKS 172 

Riveted  joints 105-138 

Boiler-shell  problem 131-138 

Countersunk  rivets ; 126 

Dimensions  of  rivet  heads 125 

Efficiencies  of 113-121,  123 

Failure  of no 

General  formulae  for 1 20 

Kinds  of 108 

Length  of  rivet 125 

Margin 122 

Materials  for 129 

Methods  of  riveting 105 

More  than  two  plates 130 

Nickel-steel  rivets 126 

Perforation  of  plate 106 

Plates  not  in  same  plane 131 

Plates  with  upset  edges 1 29 

Slippage  of 1 24 

Strength  of  materials  used  in 111-113,  126,  129 

Strength,  proportions  and  efficiency  of 113-123 

Tightness  of 122,  124,  127 

Riveting-machine,  action  of 105 

Frame  design  for 473 

ROBINSON,  PROF.  S.  W 369 

Roller-  and  ball-bearings 261-273 

Allowable  loading  for 269 

Binding,  prevention  of 273 

Efficiency  of 273 

General  considerations 261 

Lubrication  and  sealing  of 272 

Races  for 261,  268 

Rolling,  sliding,  and  spinning 261 

Size  of 271 

Rope-transmission,  fibrous 310 


496  INDEX. 

PAGE 

Rope-transmission,  wire ji8 

Rupture,  modulus  of gg 

Safety,  factor  of 82 

of  machine  operators xii 

SAMES,  C.  M I03 

Screws  and  screw  threads.     See  Bolts  and  screws. 

Section  modulus,  tabe  of  plane  and  polar 100- 101 

Set-screws 140,  177 

Shafting,  angular  distortion  of 200,  202 

Combined  thrust  and  torsion 200 

Combined  torsion  and  bending ig8 

Critical  speed  of 204 

Hollow  VS.  SOlid IQQ 

Line-shafts 201 

Simple  torsion i97 

Shaping-machine,  force  problem 64 

Quick-return  mechanism  for 37 

Sheave-wheels,  for  fibrous  ropes 312 

for  wire  ropes 320,  322 

See  also  Fly-wheels. 

SHIBATA,  M 125 

Shrink-fits 1 78-183 

SIBLEY  COLLEGE  LABORATORIES 158 

Sibley  Journal  of  Engineering 1 20 

Skew  bevel-gears 389 

Slider-crank  chain,  acceleration  diagrams 68-75,  79 

Description  of 16-17 

Force  problem,  shaping-machine 64-66,  75 

Force  problem,  steam  engine 66-74,  76-79 

Location  of  centres 16-17 

Maximum  velocity  of  slider 69,  477-483 

Tangential  effort  diagrams 76-78 

Velocity  diagram 26,  28,  79 

Sliding  pair 14 

Sliding  surfaces 185-193 

Allowable  bearing  pressure  for 191 

Form  of  guides  of 187 

General  discussion  of 185 

Lubrication  of 190 

Proportions  dictated  by  wear 185,  193 

Slotted  cross-head 20-21 

Slotting-machine,  frame  design  for 447 

Gearing  for 380 

SMITH,  C.  A 293 


INDEX.  497 

PAGE 

SMITH,  OBERLIN 230 

SMITH'S  Materials  of  Machines 8r 

SOUTHER,  H 280 

Specific  gravity  of  metals 98 

Spheres,  stresses  in 103 

Spindles.     See  Axles,  shafts,  and  spindles. 

Spiral  gears 390-41 2 

Axes  at  90° 390 

Axes  at  any  angle,  /3 405 

Splines.     See  Keys. 

Springs 429~435 

Axial  or  helical 433 

Cantilever 430 

Coil  or  spiral. . . . '. 434 

Elliptic  and  semi-elliptic 432 

Flat , 430 

Leaf 431 

Materials  and  stresses 434 

Torsioaal 434 

Spur-gears 347-380,  421 

Square  plates,  stresses  in : 104 

Stayed  surfaces,  stresses  in , 104 

Steam-engine,  boxes 249 

Crank-pin 232,  236 

Cross-head  pin 237 

Fly-wheel  design 336 

Force  problem 66-74,  76-79 

Frame,  center  crank 452 

,  «  girder  bed  » 449 

,  "  heavy  duty  " 451 

Main  journal fc-. 233 

Steam-hammer,  double-acting,  frame  design 459 

Single-acting,  frame  design 455 

Steels,  properties  of  various 88,  98,  99,  112,  43  \ 

STODOLA'S  Steam  Turbines 338 

STONEY'S  Strength  and  Proportions  of  Riveted  Joints 107,  in,  118,  123 

Straight-line  motions. ...    41-50 

General  methods  of  design 44-48 

Grasshopper 43 

Parallelogram 41 

PEAUCELLIER 48 

WATT 41 

Strength  and  stiffness  in  design ix 

See  also  Stresses. 

Stress  and  strain  formulae , ,  t T , . ,  t , , , , 102-104 


498  INDEX. 

PAGE 

Stresses  in  machine  parts,  combined 198,  200 

Compression 89 

Constant 82 

Flexure 90 

Shock  or  suddenly  applied 87 

Tables  of 102-104 

Tension  of 89 

Torsion 91 

Variable 82-87 

STRIBECK,  PROF.  C 217,  218,  219,  220,  223,  224,  269,  273,  41^  417 

Stud,  definition  of 139 

Supports 436-446 

Divided 437 

General  laws  for  design  of 436 

Reduced  number  of 439 

Three-point 438 

See  also  Brackets;  and  Frames. 

SWEET,  PROF.  JOHN  E v,  81,  94.  158,  187,  439,  440 

TALBOT,  PROF.  A.  N. 127 

TAYLOR,  FREDERICK  W 301 

THOMAS,  PROF.  CARL 273 

THOMAS'S  Worm  Gearing 390 

Thrust- journals 238-247 

THURSTON,  PROF.  R.  H 219 

Toothed  wheels  or  gears 347-428 

Addendum 349,  359 

Angle  of  action 356 

Annular ' 362 

Arc  of  action 356,  374,  375 

Backlash 349 

Bevel-gears 382 

Bronze 379 

Cast-iron 374,  375,  379 

Circular-pitch 349 

Clearance 349 

Cycloidal  teeth 353~358,  360,  373,  383 

Depth,  total 349 

Depth,  working 349 

Diametral  pitch 349 

Efficiency 379 

Elliptic 380 

Epicyclic  trains 424 

Forms  of  teeth 350,  366 

Friction,  pressure  and  abrasion ,,.,.,.,,..,,,.,., 378 


INDEX.  499 

PAGE 

Toothed  wheels  or  gears  (continued). 

Hard  fiber 376 

Helix  angle 390 

Interchangeable  sets 354,  363,  365 

Interference 359 

Involute  teeth 358,  361,  373,  383 

Line  of  pressure 356 

Non-circular  wheels 380 

Pinion 360 

Pitch  arc 356,  374,  375 

Pitch  circle ; 348 

Proportions  of 369 

Racks 360 

Rawhide 376,  379 

Reverted  trains 423 

Skew-bevel 389 

Spiral. . 390 

Spur  gear-chains 421 

Spur  wheels 347-380 

Steel 376,  379 

Step,  twisted  or  herring-bone 381 

Strength  of  teeth 371 

Stub-tooth 375 

Theory  of 347 

Twisted  bevel 388 

Velocity  coefficients 374,  375 

Worms  and  wheels 412 

Torque  diagrams 76-78 

TOWER,  BEAUCHAMP 217,  219,  240,  241,  242,  243,  252,  253,  254,  255 

Traction  and  Transmission 219 

Transactions,  Amer.  Inst.  Min.  Engs 200 

Transactions,  Amer.  Soc.  Mech.  Engs 142,  161,  177,  179,  183,  191, 

210,  213,  230,  232,  242,  254,  271,  273,  280,  293,  301,  316,  336,  338, 

343,  344,  374,  393,  4i?,  435 

Transmission,  machinery  of 4 

See  also  Belts;   Ropes;  Shafting;  and  Toothed  Wheels. 

Turning  pair 14 

Twisting  pair 14 

UNWIN'S  Machine  Design 84,  87,  130,  191,  219,  413 

Value  of  metals,  approximate 98 

Vector  quantity 25 

Velocity,  angular 24 

Definition  of 24 


5oo  INDEX. 

PAGE 

Velocity,  angular  (continued). 

Diagrams 26,  28,  79 

Linear 24 

Relative 24,  26,  28 

Watertown  Arsenal,  Tests  of  Metals 107,  in,  123 

WATT  parallel  motion 41 

Ways.    See  Sliding  Surfaces. 

WEAVER,  S.  H 205 

Weight  of  Metals 98 

WEISBACH'S  Mechanics  of  Materials 50 

WEYRAUCH,  J 83 

WHITE,  MAUNSEL 107 

WHITWORTH  quick-return  mechanism 37 

WILLIS'S  Elements  of  Mechanism 369 

WOHLER,  A 83 

Work,  definition  of i 

Worm-gearing 412 

Wrought- iron 88,  98,  99 

Zeitschrift  des  Vereins  deutscher  Ingenieure. .   217,  219,  269,  273,  378,  379,  416,  417 
Zeitschrift  fur  Math,  und  Physik 246,  247 


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